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SEMI-INFINITE COHOMOLOGY AND THE LINKAGE PRINCIPLE FOR

W-ALGEBRAS

GURBIR DHILLON

In memory of NB

Abstract Let g be a simple Lie algebra and Wκ be the affine W-algebra associated to a principalnilpotent element of g and level κ We explain a duality between the categories of smooth W modulesat levels κ+κc and minusκ+κc where κc is the critical level Their pairing amounts to a constructionof semi-infinite cohomology for the W-algebra

As an application we determine all homomorphisms between the Verma modules for W verifyinga conjecture from the conformal field theory literature of de Vosndashvan Driel Along the way wedetermine the linkage principle for Category O of the W-algebra

1 Introduction

In a celebrated work Feigin and Fuchs computed the space of intertwining operators betweenVerma modules for the Virasoro algebra [21] As a striking consequence of their calculation thefull subcategories of Verma modules at central charges c and 26 - c were opposite to one anotherFeigin had recently introduced the semi-infinite cohomology of Lie algebras into mathematics [20]For the Virasoro Lie algebra the obstruction to taking the semi-infinite cohomology of a modulevanished only at central charge 26 and it was already understood in loc cit that the dualityof Verma embeddings at complementary central charges could be recovered from properties of thesemi-infinite cohomology functor

The Virasoro vertex algebra can be realized as the quantum Hamiltonian reduction of the vacuumvertex algebra for affine sl2 Applying the same procedure for other simple Lie algebras g producesthe W-algebras A natural conjecture which appeared in the conformal field theory literature isthat a version of FeiginndashFuchs duality for Verma modules should persist in this wider setting [17]Moreover it was understood that this again should follow from having a theory of semi-infinitecohomology for W-algebras

However generalizing from sl2 to general g proved to be far from automatic Since the operatorproduct expansions of the standard generating fields of a general W-algebra are not linear in thefields unlike the case of the Virasoro vertex algebra the previous Lie algebra formalism for semi-infinite cohomology was inapplicable Nonetheless it was clear such a process should exist and playa similarly basic role eg in cancelling unphysical states against ghosts in W-gravity and W-stringtheory [13] [14] [50] Despite constructions of such a cohomology theory and their detailed studyfor other small rank examples no general method was found

In this paper we construct a semi-infinite cohomology theory for W-algebras To do so wewill make essential use of several recent breakthroughs in the local quantum geometric Langlandsprogram Having done so we will as conformal field theorists knew all along see an appropriateform of FeiginndashFuchs duality for the W-algebra Along the way we will prove several basic resultson Category O for the W-algebra notably a long expected linkage principle

Date Spring 2019

1

2 GURBIR DHILLON

2 Statement of Results

Let g be a simple complex Lie algebra with triangular decomposition g = nminus oplus hoplus n1 Let G bethe simple simply connected algebraic group with Lie algebra g Fix a level κ ie an invariantbilinear form κ isin (glowast otimes glowast)G and write κc for the critical level Write gκ for the affine Lie algebraassociated to g and κ and let Wκ denote the W-algebra associated to gκ and a principal nilpotentelement of g

21 Categorical FeiginndashFuchs duality We prove a version of FeiginndashFuchs duality which ap-plies to all representations of the W-algebra Since this uses several modern notions from homo-logical algebra which may not be familiar to all readers we will approach it via several moreelementary statements A reader comfortable with the basics of stable cocomplete infin-categoriesmay wish to skip directly to Theorem 25

As mentioned in the introduction Feigin and Fuchs found a remarkable duality between theVerma modules for Virasoro at complementary central charges ie a contravariant equivalenceof categories We show this phenomenon persists for W Let us write Wκ -modhearts for its abeliancategory of representations and Vermaheartsκ for its full subcategory consisting of Verma modules Thenwe have

Theorem 21 There is a canonical equivalence of categories

Vermaheartsopκ Vermaheartsminusκ+2κc

To our knowledge Theorem 21 is new for all cases besides the Virasoro algebra A next approx-imation to our FeiginndashFuchs duality is roughly the statement that not only Homs between Vermasmatch but also Exts Let us write Wκ -modb for its bounded derived category of representations orbetter its canonical dg-enhancement If we consider Vermaκ the full pretriangulated subcategorygenerated by Vermaheartsκ we have

Theorem 22 There is a canonical equivalence of dg-categories

Vermaopκ Vermaminusκ+2κc

Another approximation of FeiginndashFuchs duality is the statement that as Verma modules areprotypical objects of tame ramification the above admits variants with wild ramification To statethis cleanly involves a slightly subtle point which we now review Let us write Wκ -modn (n fornaive) for the usual unbounded derived category of Wκ modules As objects of Wκ -modn theVerma modules are not compact However there is a slight enlargement of Wκ -modn which wedenote by Wκ -mod in which the Verma modules are now compact An analogous renormalizationwas first introduced for gκ by FrenkelndashGaitsgory and later for Wκ by Raskin [25] [53]2 Suchrenormalizations are now more broadly seen as useful facts of life in both singular finite dimensionaland smooth infinite dimensional algebraic geometry and quantizations thereof [32] [34]

Within Wκ -mod one has its subcategory of compact objects Wκ -modc We now give a heuristicsense of what these look like Recall that Zg the center of the universal enveloping algebraof g quantizes the invariant theory quotient gG Similarly the topological associative algebraassociated to W quantizes the loop space L(gG) Bounding the order of poles of an algebraic

1All results we prove have analogues for the Lie algebra of a general reductive group which may be deduced fromthe simple case Similarly one may replace our ground field C with any algebraically closed field of characteristiczero

2As a remark for experts - we in fact show all of gκ -modn embeds fully faithfully into gκ -mod cf Proposition55 Previously this was only known for the bounded below derived category and the same proof applies to W -modas well

SEMI-INFINITE COHOMOLOGY AND THE LINKAGE PRINCIPLE FOR W-ALGEBRAS 3

loop exhibits L(gG) as an ascending union of pro-finite-dimensional affine spaces and objects ofWκ -modc quantize pushforwards of perfect complexes from these subschemes3

With these explanations in place we can state


Wκ -modcop Wminusκ+2κc -modc (24)

In the case of the Virasoro algebra it had long been a folklore conjecture that something like(24) should be true Positselski has given a solution in the interesting monograph [52] for anyTate Lie algebra His formulation and in particular the types of exotic derived categories he workswith are seemingly different from ours We also prove a Tate Lie algebra analogue of Theorem 23in Theorem 56 and it would be good to understand the precise relationship between this and loccit However for all other W-algebras we unaware of previous work along these lines

Finally we explain a reformulation of Theorem 23 that applies to the entirety of Wκ -modThe totality of all C-linear cocomplete dg-categories can be organized into an (infin 2)-categoryDGCatcont whose homotopy classes of 1-morphisms are C-linear continuous quasifunctors Thisis further a symmetric monoidal infin-category so that given two cocomplete dg-categories CD onecan form their tensor product C otimes D In particular one can make sense of dual objects CCor inDGCatcont and Theorem 23 is equivalent to

Theorem 25 There is a canonical duality of cocomplete dg-categories

Wκ -modorWminusκ+2κc -mod (26)

From a duality datum one obtains a contravariant equivalence between the compact objectsrecovering Theorem 23 and we show this sends Verma modules to Verma modules recoveringTheorems 21 22 Moreover as dual categories representations of complementary W-algebrashave a lsquoperfect pairingrsquo

Wκ -modotimesWminusκ+2κc -modrarr Vect

which is the promised semi-infinite cohomology functor The problem of constructing such a functorwas first raised in the conformal field theory and string theory literature and solved in several smallrank cases [9] [10] [11] [13] [14] [40] [41] [51] [57] [58] [59] [65] Its existence has also beenanticipated in the mathematical literature Notably I Frenkel and collaborators have developeda rich program on the relation between quantum groups and affine Lie algebras that links boththrough W A crucial role is played by remarkable conjectural calculations of W semi-infinitecohomology eg a putative construction of algebras of functions on quantum groups via themodified regular representations of W-algebras [27] [28] [61] [66] [67]

22 Homomorphisms between Verma modules and the linkage principle Feigin andFuchs not only proved a contravariant equivalence as in Theorem 21 for the Virasoro algebra butalso explicitly determined all the morphisms in the category Vermaheartsκ We will presently describe asimilar classification for a general W-algebra Recall that highest weights for Wκ ie the maximalspectrum of its Zhu algebra identify with Wfhlowast where Wf denotes the finite Weyl group cfSubsection 312 for our normalizations

Fix a noncritical level κ To describe Vermaheartsκ we first determine the block decomposition ofCategory O Ie we determine the finest partition

Wfhlowast =⊔α

Pα

3In modern parlance the latter are IndCoh(L(gG))c

4 GURBIR DHILLON

such that every module M in O decomposes as a direct sum of submodulesoplus

αMα wherein all thesimple subquotients of Mα have highest weights in Pα

Loosely we find that blocks for Wκ are projections of blocks for KacndashMoody More preciselyconsider the level κ dot action of the affine Weyl group W on hlowast and the projection

π hlowast rarrWfhlowast

For λ isin hlowast write Wλ for its integral Weyl group Wfλ for the intersection Wλ capWf and W λ forthe stabilizer of λ in Wλ

Theorem 27 (Linkage principle) The block of O containing the simple module with highest weightπ(λ) has highest weights

π(Wλ middot λ) WfλWλWλ

Theorem 27 has been anticipated as a folklore conjecture since the earliest days of the subjectcf [13] To our knowledge this is its first appearance for all cases save the Virasoro algebra Itwas also likely found by Arakawa (unpublished) Using Theorem 27 we formulate in Section 8several conjectures on the structure of blocks and a conjectural relation between DrinfeldndashSokolovreduction and translation functors

To describe the homomorphisms between Verma modules within a single block recall that sinceκ is noncritical Wλ middot λ always contains an antidominant or dominant weight Specifically if κ isnegative every block contains an antidominant weight and if κ is positive every block contains adominant one Thus we may assume that λ is (anti-)dominant in which case Wfλ and W λ areparabolic subgroups of Wλ and hence WfλWλW

λ carries a Bruhat order

Theorem 28 For w isin WfλWλWλ write Mw for the corresponding Verma module Then for

y w isinWfλWλWλ we have in the abelian category Wκ -modhearts

(1) Hom(MyMw) is at most one dimensional and any nonzero morphism is an embedding(2) If λ is antidominant then Hom(MyMw) is nonzero if and only if y 6 w(3) Suppose λ is dominant Then Hom(MyMw) is nonzero if and only if y gt w

Theorem 28 was conjectured in the conformal field theory literature by de Vostndashvan Driel inthe remarkable paper [17] The main non-elementary input into Theorem 27 and the negativelevel cases of Theorem 28 is Arakawarsquos resolution of the FrenkelndashKacndashWakimoto conjecture onDrinfeldndashSokolov reduction [2] [26] It is unclear whether Theorem 28 at positive level can beproven by similar methods However following the fundamental insight of Gaitsgory and hisschool that categorical dualities can see concrete representation-theoretic phenomena we insteaduse Categorical FeiginndashFuchs duality to reduce to the negative level case

23 Future directions The constructions and applications described in this paper mostly con-cern the algebraic representation theory of W-algebras However we would like to mention two fur-ther places where its contents are relevant that have a slightly different flavor namely the geometricrepresentation theory of W-algebras and applications to the quantum Langlands correspondence

First a basic problem still unsolved for the Virasoro algebra has been to produce a localizationtheorem for highest weight representations of W-algebras analogous to those available for simple andaffine Lie algebras As we will explain elsewhere this can be done using Whittaker sheaves on theenhanced affine flag variety Moreover to explicitly identify the representations realized in this wayone needs to speak of blocks of Category O for the W-algebra This motivated our determinationof the linkage principle in the current work Further at positive level the localization is essentially


of derived nature4 This motivated the present study of Categorical FeiginndashFuchs duality whichmay be used to reduce to the t-exact negative level case

Second the closely related category of Whittaker sheaves on the affine Grassmannian is the sub-ject of the conjectural Fundamental Local Equivalence (FLE) of the quantum geometric Langlandsprogram [30] [31] This conjecture due to GaitsgoryndashLurie provides a remarkable deformationof the Geometric Satake isomorphism to all KacndashMoody levels Conjectures of AganacicndashFrenkelndashOkounkov suggest that the FLE should be provable using the representation theory of W-algebras[1] We can currently give such a proof of the FLE over a point ie non-factorizably for g = sl2and the case of general g over a point is work in progress with Raskin The above localization theo-rem and the results of this paper play a basic role Similar arguments which are work in progressapply to a tamely ramified variant of the FLE conjectured by Gaitsgory [35] for appropriatelyintegral levels this has been implemented in a forthcoming work with Campbell [16]

24 Organization of the Paper In Section 3 we collect basic definitions and notations InSection 4 as preparation for FeiginndashFuchs duality we prove a general duality statement for Whit-taker models of categorical loop group representations In Section 5 we derive FeiginndashFuchs dualityfor W-algebras and Tate Lie algebras In Section 6 we prove some basic structural properties ofCategory O for Wκ In Sections 7 and 8 we prove the linkage principle and propose a conjecturaldescription of the blocks for the W-algebra Finally in Section 9 we prove de Vostndashvan Drielrsquosconjecture on Verma embeddings

Acknowledgments It is a pleasure to thank Dima Arinkin Roman Bezrukavnikov AlexanderBraverman Dan Bump Thomas Creutzig Davide Gaiotto Dennis Gaitsgory Alexander Gon-charov Edward Frenkel Igor Frenkel Flor Hunziker Victor Kac Sam Raskin Ben WebsterEmilie Wiesner David Yang Zhiwei Yun and Gregg Zuckerman for helpful discussions and corre-spondence

3 Preliminaries

In this section we establish notation and collect some facts which will be useful to us The readermay wish to skip to the next section and refer back only as needed

Our notation and conventions are standard with the mild exceptions of our normalizations ofthe isomorphism Zhu(Wκ) Zg and the duality on Category O for Wκ cf Remark 312 andSubsection 37 respectively

31 The affine KacndashMoody algebra Recall that g = nminus oplus h oplus n is a simple Lie algebra LetgAKM = g[z zminus1] oplus Cc oplus CD be the affine KacndashMoody algebra associated to g cf [43] We usethe standard normalizations so that

[X otimes zn Y otimes zm] = [XY ]otimes zn+m + nδnminusmκb(XY )c X Y isin g nm isin Z

where κb is the basic invariant inner product ie normalized so that κb(θ θ) = 2 where θ isin h isthe coroot associated with the highest root θ isin hlowast of g The remaining brackets are

[Cc gAKM ] = 0 [DX otimes zn] = nX otimes zn X isin g

The affine KacndashMoody algebra has a standard triangular decomposition g = nminus oplus h oplus n whereh = h oplus Cc oplus CD Write αi isin h αi isin hlowast i isin I for the simple coroots and roots of g Then thesimple coroots and roots for gAKM are naturally indexed by I = I t 0 Explicitly the simplecoroots are given by

αi isin hoplus 0oplus 0 i isin I and α0 = minusθ + c isin hoplus Ccoplus 0

4In particular a version of KashiwarandashTanisaki localization which has not yet appeared in the literature holdsfor gκ at positive level

6 GURBIR DHILLON

To write down the simple roots we decompose

hlowast hlowast oplus (Cc)lowast oplus (CD)lowast = hlowast oplus Cclowast oplus CDlowast

where 〈clowast c〉 = 1 〈Dlowast D〉 = 1 With this the simple roots are given by

αi isin hlowast oplus 0oplus 0 i isin I and α0 = minusθ +Dlowast isin hlowast oplus 0oplus CDlowast

32 The affine Weyl group and the level κ action For each index i isin I we have an associatedsimple reflection

si hlowast rarr hlowast si(λ) = λminus 〈λ αi〉αi

The subgroup of GL(hlowast) generated by these reflections si i isin I is the affine Weyl group W Thesubgroup generated by si i isin I is the finite Weyl group Wf associated to g In both cases thesepreferred generators exhibit W and Wf as Coxeter groups

By construction W acts trivially on CDlowast It follows that W acts linearly on the quotienthlowastCDlowast hlowast oplusCclowast The action of W on hlowast oplusCclowast further preserves the affine hyperplanes Hk cutout by c = k k isin C5 Therefore under the affine linear isomorphism hlowast Hk sending λ isin hlowast toλ + kclowast we obtain an action of W on h by affine linear automorphisms which we call the level kaction

Explicitly at level k Wf acts in the usual way on hlowast and s0 acts by the affine reflection throughθ = k

s0(λ) = λminus (〈λ θ〉 minus k)θ

By composing s0 with the reflection sθ isinWf associated with root θ one obtains translation by kθUsing this one finds

Proposition 31 Write Λ sub hlowast for the lattice generated by the long roots Then the level k actionof W on hlowast identifies with Wf nΛ where Wf acts by usual reflections and λ isin Λ acts by the dilatedtranslation tkλ(ν) = ν + kλ ν isin hlowast

33 The affine Lie algebra at level κ and the level κ action Recall that to any invariantinner product κ on g we have associated central extension of g((z))

0rarr C1rarr gκ rarr g((z))rarr 0

with commutator given by

[X otimes f Y otimes g] = [XY ]otimes fg + κ(XY ) Resz=0 fdg X Y isin g f g isin C((z))

Since g is simple we may write κ = kκb k isin CWrite gAKM -modheartsk for the abelian category of smooth representations of gAKM on which c acts

by the scalar k isin C Similarly write gκ -modhearts for the abelian category of smooth representations ofgκ on which 1 acts by the identity By smoothness restricting an action of gAKM to g[z zminus1]oplusCcproduces a functor gAKM -modheartsk rarr gκ -modhearts When κ 6= κc this functor admits a distinguishedsection given by the SegalndashSugawara construction cf [23] With these identifications we have

Proposition 32 Let κ = kκb κ 6= κc Then the level k action of W on hlowast considered in Subsection32 coincides with the standard action of W on the weights of h-integrable gκ modules

Henceforth we will speak interchangeably of the level k and κ actions

5Of course the same holds without quotienting by CDlowast


34 Combinatorial duality Having explained the level κ action we presently discuss a combi-natorial shadow of the duality between gκ -mod and gminusκ+2κc -mod From Proposition 31 W actsat level κ through Wf n Λ where the translation action of Λ is dilated by κ

κb It follows that the

orbits for the level κ and κprime actions coincide only when κ + κprime = 0 In this case it is not quitethe case that level κ and κprime actions coincide since s0 acts as the affine reflection through θ = plusmn κ

κb

Incorporating this sign we have

Proposition 33 Write hlowastκnaive for hlowast with the level κ action of W Then negation λrarr minusλ is aW-equivariant isomorphism hlowastκnaive hlowastminusκnaive

The appearance of the critical twist comes as usual from a ρ shift To explain this writeρ isin hlowast for the unique element satisfying 〈ρ αi〉 = 1 i isin I Write ρ isin hlowast for an element satisfying

〈ρ αi〉 = 1 i isin I Explicitly the possible ρ form the affine line ρ+horclowast+CDlowast where hor is the dualCoxeter number Recalling that κc = minushorκb in particular any ρ will lie in the affine hyperplane ofhlowast corresponding to minus the critical level

We then have the dot action of W on hlowast where w middot λ = w(λ + ρ) minus ρ which is independent ofthe choice of ρ This is the usual action albeit centered at minusρ rather than 0 That is the maphlowast rarr hlowast sending λ to λminus ρ intertwines the usual action of W on the domain and the dot action ofW on the target

As in Subsection 32 the dot action descends to an action on hlowast oplus Cclowast This action preserveseach affine hyperplane c = κ

κb which gives us a level κ dot action on hlowast hlowast + κ

κbclowast sub hlowast oplus Cclowast

which we denote by hlowastκ By construction we have an isomorphism

ι hlowastκnaive hlowastκ+κc ι(λ) = λminus ρ λ isin hlowastκnaive (34)

Concatenating Equation (34) and Proposition 33 gives

Proposition 35 There is a W equivariant isomorphism

i hlowastκ hlowastminusκ+2κc i(λ) = minusλminus 2ρ λ isin hlowastκ (36)

As we will see this naive duality between orbits at positive and negative levels will show up inthe combinatorics of the tamely ramified cases of KacndashMoody duality in particular the duality ofVerma modules cf Theorem 95

35 The W-algebra Some nice references for W-algebras are [3] [24] Recall we wrote g =nminus oplus h oplus n For a complex Lie algebra a write La for the topological Lie algebra a((z)) andsimilarly L+a = a[[z]]

Fix a nondegenerate character ψ nrarr C ie one which is nonzero on each simple root subspaceof n Still write ψ for the character of Ln given by the composition

LnResminusminusrarr n

ψminusrarr C

which depends on our choice of coordinate z Writing Cψ for the associated character representationof Ln and Vκ for the vacuum algebra for gκ cf [23] [29] we have

Wκ = Hinfin2

+0(Ln L+nVκ otimes Cψ) (37)

Wκ is a vertex algebra and for κ noncritical contains a canonical conformal vector ω Write L0 forthe corresponding energy operator ie the coefficient of zminus2 in the field ω(z) Under the adjointaction of L0 Wκ acquires a Zgt0 grading We say a isin Wκ is homogeneous of degree |a| isin Zgt0 ifL0a = |a|a

8 GURBIR DHILLON

36 Category O In what follows we only consider levels κ 6= κc Consider the abelian categoryWκ -modhearts of modules for Wκ cf [24] The following definition is more or less due to Arakawa in[2] see however Remark 310

Definition 38 O is the full subcategory of W -modhearts consisting of objects M satisfying

(1) M decomposes as a sum of generalized eigenspaces under the action of L0 ie we maywrite

M =oplusdisinC

Md Md = m isinM (L0 + d)Nm = 0 N 0

(2) For each d isin C Md is finite dimensional and Md+n = 0 for all n isin ZgtN N 0

Remark 39 Allowing generalized eigenspaces of L0 may be compared with allowing generalizedeigenvalues of the Cartan subalgebra and generalized central characters in the formation of CategoryO for a semisimple Lie algebra ie we obtain the lsquofree-monodromicrsquo Category O To our knowledgea good definition of the lsquonon-monodromicrsquo category for W-algebras other than Virasoro is unknown

Remark 310 We should mention that since κ 6= κc we do not consider the grading as an auxiliarystructure but rather induced from the conformal vector of W This differs from the treatment ofArakawa [2] as in the discussion of Subsection 33 for KacndashMoody

LetM be an object of O Form isinM recall thatm is said to be singular if for every homogeneousa isinWκ a(z)m isinM((z)) has a pole of at most order |a| Write S(M) for the subspace of all singularvectors in M This naturally carries an action of the Zhu algebra Zhu(Wκ) The latter can beidentified with Zg the center of the universal enveloping algebra of g We will explain our precisenormalization for this isomorphism in Remark 312 below but in particular by the HarishndashChandraisomorphism we may identify characters of Zhu(Wκ) withWfhlowast the quotient of hlowast by the dot actionof Wf

If m isin S(M) is an eigenvector for the action of Zhu(W) with eigenvalue λ isin Wfhlowast we saym is a highest weight vector with highest weight λ Morphisms in O preserve singular and highestweight vectors Accordingly for λ isinWfhlowast there is an associated functor mλ Orarr Vect sendingan object M to its highest weight vectors of weight λ This is corepresented by an object M(λ)called the Verma module M(λ) has a unique simple quotient L(λ) and every simple object of Ois of the form L(λ) for a unique λ isinWfhlowast

We say a module M is a highest weight module if it can be generated by a highest weight vectorEquivalently M admits a surjection from M(λ) for some necessarily unique λ isinWfhlowast

Feeding representations of gκ -modhearts into a complex similar to Equation (37) produces repre-sentations of Wκ The following fundamental theorem of Arakawa informally says that this relatesCategory O for gκ and Wκ in an extremely tight way

Theorem 311 [2] Write Ogκ for the Category O associated to gκ For λ isin hlowast write M(λ) andL(λ) for the associated Verma and simple modules for gκ and recall the projection π hlowast rarrWfhlowastThere exists an exact functor

DSminus Ogκ rarr O

such that for any λ isin hlowast DSminus(M(λ)) M(π(λ)) and

DSminus(L(λ))

L(π(λ)) 〈λ+ ρ αi〉 isin Zgt0 i isin I0 otherwise

Remark 312 In particular we normalize the isomorphism Zhu(Wκ) Zg by composing the iso-morphism Zhu(Wκ) Zg used by Arakawa in loc cit with the involution Zg Zg correspondingto the automorphism of Wfhlowast given by π(λ) 7rarr π(minuswλ) where w denotes the longest elementof Wf


37 Duality In loc cit Arakawa shows that O admits a contravariant equivalence Dprime Orarr OopBy definition the underlying vector space of DprimeM is

oplusdisinCM

lowastd and the action of W is given by

the adjoint action precomposed by an appropriate Chevalley anti-involutionIn loc cit it is shown that DprimeL(π(λ)) L(π(minuswλ)) We may modify it to remove the

appearance of w as follows Fix Chevalley generators ei hi fi i isin I for g The longest element wof Wf sends the positive simple roots to the simple negative roots Ie we obtain an involution τof I such that

Adw Cei = Cfτ(i) i isin IWrite τ for the involution of g sending ei hi fi to eτ(i) hτ(i) fτ(i) i isin I respectively This induces

involutions τ of L+g and gκ and hence Vk Similarly it induces involutions of Ln and L+n andif we pick ψ so that ψ(ei) = 1 i isin I we obtain an involution τ of C

infin2

+lowast(Ln L+nVk otimes Cψ) and

hence of Wκ Writing τlowast for the restriction functor Wκ -modhearts rarr Wκ -modhearts induced by τ wedefine D = τlowast Dprime With this we have

DL(λ) L(λ) λ isinWfhlowastLet us say that the module A is co-highest weight if DA is a highest weight module Calling

A(λ) = DM(λ) a co-Verma module equivalently a module A is co-highest weight if it admits aninjection into a necessarily unique DM(λ) for some λ isinWfhlowast

38 q-characters The following will only be used in the proof of Lemma 74 For an objectM =

oplusdisinCMd of O its q-character is the formal series

chqM =sumdisinC

(dimMminusd)qd

The basic properties which will be of use to us are summarized in the following

Proposition 313 For any object M of O we have

chqM = chq DM

In particular chqM(λ) = chq A(λ) λ isinWfhlowast

4 Duality for Whittaker models of categorical loop group representations

In this section we use the adolescent Whittaker filtration of Raskin to construct a duality forWhittaker models of strong categorical representations of loop groups A very readable reference forbackground on categorical representations of groups and loop groups and in particular D-moduleson loop groups is [6]

Let us mention that this duality is already known to experts Namely the commutation of Whit-taker invariants with tensoring by an object of DGCatcont viewed as a trivial strong representationalready formally implies the duality of Whittaker models This in turn follows from knowing thatWhittaker invariants and coinvariants can be identified cf [8] [36] However the present ado-lescent Whittaker construction of the duality which has not yet appeared in the literature givesa transparent picture helpful for concrete applications in particular the relation to dualities onspherical and baby Whittaker invariants and the action on compact objects These compatibilitiesare crucial for representation-theoretic applications in the remainder of the paper

41 Recollections on the adolescent Whittaker construction Let U be a quasicompactpro-unipotent group scheme and C a cocomplete dg-category with a strong action of U Recallthat we may form the invariants CU and the coinvariants CU and that the tautological composition

CUOblvminusminusminusrarr C

insminusminusrarr CU (41)

gives an equivalence CU CU [7] Recall that G was simple with Lie algebra g = nminus oplus h oplus nWrite N for the connected unipotent subgroup of G with Lie algebra n and L+GL+N for their

10 GURBIR DHILLON

arc groups and LGLN for their loop groups respectively Let C be a cocomplete dg-categorywith a strong action of LN LN is now an ind-prounipotent group scheme and the analogous mapCLN rarr CLN need not be an equivalence However a theorem of Raskin salvages a twisted versionof this statement as we now remind For ψ LN rarr Ga a character we may form the invariantsCLNψ and coinvariants CLNψ

The relevant ψ is obtained as follows Let ψ N rarr Ga be a character Then ψ necessarilyfactors through N[NN ] For each simple root αi i isin I we have the corresponding one parametersubgroup Uαi rarr N and an isomorphism

N[NN ] prodiisinI

Uαi

Recall that ψ is said to be nondegerate if its restriction to each simple root subgroup Uα is nonzeroAssociated to our choice of coordinate z is a residue map Res LGa rarr Ga Accordingly we mayform the composition

LN rarr L(N[NN ])Resminusminusrarr N[NN ]

ψminusrarr Ga (42)

which we continue to denote by ψ

Theorem 43 [53] Let ψ LN rarr Ga be a character as in Equation (42) Then for any κ and C

with a strong level κ action of LG there is a canonical isomorphism

CLNψ CLNψ (44)

We emphasize that one asks that the action of LN be the restriction of an action of LG andindeed the functor realizing the equivalence of Theorem 43 is not the analog of (41) Instead aswe recall presently it is subtler and uses more of the LG action

Consider the decreasing sequence of subgroups of L+G

I primen = L+GG[z]zn

times N [z]zn n isin Zgt1

In words I primen is jets into G which to (n minus 1)st order lie in N Note that for all n gt 1 I primen isprounipotent Raskin skews the I primen by the cocharacter ρ of the adjoint torus Had to obtain

In = Adtminusnρ In n isin Zgt1

Using the map I primen rarr N [z]zn ψ induces a homomorphism In rarr Ga which coincides with (43) onIn capN(K) Accordingly one can form the adolescent Whittaker categories CInψ n gt 1

Due to conjugation by tminusnρ the In no longer form a decreasing series of subgroups Instead forn 6 m one relates the adolescent Whittaker categories by the functors

inmlowast CInψOblvminusminusminusrarr CIncapImψ

Avlowastψminusminusminusminusrarr CImψ

inm CImψOblvminusminusminusrarr CIncapImψ

Avlowastψminusminusminusminusrarr CInψ

Informally inmlowast averages an element of C by Im cap LNIn cap LN and inm averages an element of

C by In cap LBminusIm cap LBminus Raskin proves that inm admit fully faithful left adjoints inm A keyresult is that this coincides with inmlowast up to a shift

inm inmlowast[2(mminus n)∆] (45)

where ∆ = dim AdtminusρL+NL+N

Raskin moreover observes that

CLNψ limminusrarrinmlowast

CInψ CLNψ limminusrarrinm

CInψ


Here and elsewhere the above homotopy colimits and limits are taken in the setting of cocompletedg-categories unless otherwise specified By virtue of (45) the maps id[minus2m∆] CInψ rarr CInψ

yield the desired isomorphism CLNψ CLNψ of (43)

42 Whittaker invariants of dual representations prounipotent case Since we are un-aware of a reference with proofs for the desired duality and its basic properties in the prounipotentcase though it is well known to experts we first write this down in some detail

421 Dualizability Suppose U is a quasicompact prounipotent group scheme We remind theisomorphism between invariants and coinvariants in more detail Let C be a category stronglyacted on by U Then the invariants category CU comes equipped with an adjunction of continuousfunctors

Oblv CU C Avlowast (46)

wherein Oblv is fully faithful Similarly the coinvariants category CU comes equipped with anadjunction of continuous functors

insL CU C ins (47)

wherein insL is fully faithful The following assertion may be found in [7]

Theorem 48 The composition ins Oblv CU rarr CU is an equivalence

The following compatibility will be useful to us Throughout the paper we understand commu-tative diagrams of dg-categories to commute up to natural equivalence

Proposition 49 The equivalence of Theorem 48 exchanges the adjunctions (46) (47) ie fitsinto commutative diagrams

CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins

Proof In loc cit an inverse functor ι to ins Oblv is constructed by applying the universal propertyof CU to Av C rarr CU This yields the right hand triangle and the other is obtained by passingto left adjoints

We can now state how dualizability interacts with invariants and coinvariants for prounipotentgroups Recall that DGCatcont denotes the (infin 2)-category of cocomplete C-linear dg-categoriesand continuous C-linear quasi-functors between them equipped with the Lurie tensor product cf[33] Let C be a dualizable object of DGCatcont with dual Cor If C is equipped with a strong actionof U then Cor acquires a strong right action of U by the dualizability of D(U) Applying inversionon the group one converts this into a strong left action of U

Proposition 410 Let C Cor be as in the preceding paragraph Then CU is again dualizable withdual (Cor)U and pairing

CU otimes (Cor)UOblvotimesOblvminusminusminusminusminusminusminusrarr Cotimes Cor rarr Vect (411)

Proof We first show the dualizability of CU CU For an arbitrary test object S of DGCatcont wemay equip it with a trivial strong action of U and compute

HomDGCatcont(CU S) HomD(U) -mod(C S) HomDGCatcont(C S)U (Cor otimes S)U

To show dualizability it remains to show the natural map (Cor)U otimesSrarr (CorotimesS)U is an equivalenceie we would like to pull the S out of the invariants Since the latter is a totalization and in

12 GURBIR DHILLON

particular an inverse limit this is not completely formal However the analogous assertion forcoinvariants is and so we compute using Theorem 48

(Cor otimes S)U (Cor otimes S)U (Cor)U otimes S (Cor)U otimes S

By construction the perfect pairing δ CU otimes (Cor)U rarr Vect fits into a commutative diagram

Cotimes (Cor)UidotimesOblv

insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99

Precomposing the diagram with insLotimes id and using the fully-faithfulness of insL we deduce thepairing can be rewritten as

CU otimes (Cor)UinsLotimesOblvminusminusminusminusminusminusminusrarr Cotimes Cor rarr Vect

By Theorem 48 and Proposition 49 we can replace CU and insL with CU and Oblv as desired

Recall that to a continuous functor F Crarr D between dualizable objects of DGCatcont one canassociate a dual functor For Dor rarr Cor We now determine the effect of duality upon the forgetfuland averaging functors namely that they swap

Proposition 412 Under the equivalence (CU )or (Cor)U of Proposition 410 the functors

Oblv CU C Av

are dual to the functorsOblv (Cor)U Cor Av

ie there are natural isomorphisms Oblvor Av Avor Oblv

Proof For a dualizable object D of DGCatcont with dual Dor let us write 〈minusminus〉D isin Vect forthe pairing D otimes Dor rarr Vect To prove the Proposition it suffices to provide an equivalence ofquasi-functors

〈minusAvlowastminus〉CU 〈Oblvminusminus〉CBy the definition of the pairing between CU (Cor)U we have

〈minusAvlowastminus〉CU 〈OblvminusOblv Avlowastminus〉CRecall that Oblv Av is given by convolution by kU the constant sheaf on U cf [7] Writinginv U rarr U for inversion by the definition of the action of D(U) on Cor we have

〈OblvminusOblv Avlowastminus〉C 〈Oblvminus kU minus〉C 〈invlowastdR kU Oblvminusminus〉CAs invlowastdR kU kU the fully-faithfulness of Oblv yields

〈kU Oblvminusminus〉C 〈Oblvminusminus〉Cas desired Having shown the duality of Oblv CU rarr C and Avlowast Cor rarr (Cor)U the other claimeither follows from the involutivity of taking dual categories

Finally we will need a Whittaker twist of the above So suppose χ U rarr Ga is a characterAssociated to this is a character representation Vectχ Let us recall the construction Write z forthe standard coordinate on Ga and D for the algebra of global differential operators on Ga Thenone has the exponential (left) D-module

ez = DD(partz minus 1)[1] (413)

Writing eχ for χez then eχ is a character D-module on U ie is equipped with an isomorphismmeχ eχeχ satisfying an associativity compatibility where m UtimesU rarr U is the multiplication


map Accordingly one has a representation Vectχ of D(U) whose underlying dg-category is Vectand whose coaction

Vecta

minusrarr VectotimesD(U) D(U)

sends V isin Vect to V otimes eχ Note that when χ = 0 eχ is the dualizing sheaf ωU and this recoversthe usual trivial representation of D(U)

Let us recall that these multiply in the expected in way

Lemma 414 Let χ χprime be characters of U Then there is a canonical isomorphism of strong rep-resentations VectχotimesVectχprime Vectχ+χprime In particular we have a D(U)-equivariant duality betweenVectχ and Vectminusχ

Proof This follows from the isomorphism of character D-modules eχotimes eχprime eχ+χprime

For a strong representation C of D(U) and an additive character χ let us write C(χ) for therepresentation CotimesVectχ With this we have by Lemma 414

CUχ = HomD(U) -mod(VectχC) HomD(U) -mod(VectC(minusχ)) = C(minusχ)U

In particular as before there is an adjunction

Oblvχ CUχ C Avχlowast

wherein Oblvχ is fully faithful Let us now record the interaction of dualizability with twistedinvariants

Proposition 415 Let C be a strong representation of U and χ U rarr Ga a character Then if Cis dualizable then the pairing

CUχ otimes (Cor)UminusχOblvotimesOblvminusminusminusminusminusminusminusrarr Cotimes Cor rarr Vect

gives an equivalence CUχor (Cor)UminusχWith respect to this duality datum we have equivalences offunctors

Oblvorχ Avminusχlowast Avorχlowast Oblvminusχ

Proof We compute

(CUχ)or (C(minusχ)U )or ((C(minusχ))or)U (Cor(χ))U (Cor)Uminusχ

where in the middle we used Proposition 410 The remaining assertions can be deduced fromapplying Propositions 410 and 412 to C(minusχ)

422 Dualizability and compact objects generalities Let D be a cocomplete dg-category Recallan object d of D is compact if Hom(dminus) Drarr Vect commutes with direct sums Writing Dc forthe subcategory of compact objects we obtain a map

Dcop rarr Hom(DVect)

Then we have

Proposition 416 For d as above dor is a compact object of Hom(DVect) Moreover if D isdualizable then using the canonical identification

Hom(DVect) Hom(VectDor) Dor

the assignment drarr dor yields an equivalence

D Dcop Dorc (417)

14 GURBIR DHILLON

Proof Let us write momentarily HomnaiveDGCat(DVect) for the dg-category of bona fide dg-functors

from D to Vect localized at quasi-isomorphisms Let us similarly writeHomnaive

DGCatcont(DVect) for the full subcategory of continuous dg-functors If we write HomDGCat(DVect)for the inner Hom between D and Vect in DGCat ie the dg-category of quasi-functors then thereis a natural map

HomnaiveDGCat(DVect)rarr HomDGCat(DVect) (418)

It is known that Equation (418) is an equivalence cf [62] Recalling that Hom in DGCatcont isthe full subcategory of HomDGCat(DVect) consisting of continuous quasi-functors it follows thatEquation (418) induces an equivalence

HomnaiveDGCatcont(DVect) Hom(DVect) (419)

The upshot is that there is no distinction between quasi-functors and functors to Vect ie we maywork naively Recall by the dg-Yoneda lemma we have for d isin D and ξ isin Homnaive

DGCat(DVect)evaluation on d yields an equivalence

Hom(dor ξ)simminusrarr 〈d ξ〉

where 〈d ξ〉 isin Vect denotes the evaluation of ξ on d It follows that any dor is a compact object ofHomnaive

DGCat(DVect) as sums of functors are computed lsquopoint-wisersquo In particular if d is compactthen dor is a compact object of Hom(DVect) Applying dg-Yoneda therefore yields a fully-faithfulembedding Dcop rarr Dorc The involutivity of duality provides a reverse map Dorc rarr Dcop andthese are readily seen to be inverse equivalences

Remark 420 This recovers the well known fact that if D is compactly generated then D isdualizable with dual Ind(Dcop) cf [38] However we do not know a reference for (417) in thenon-compactly-generated case

423 Dualizability and compact objects prounipotent case Let us see how compact objects trans-form under the functors discussed in Section 42 In particular U is still a quasicompact prounipo-tent group χ U rarr Ga is a character and C is a strong representation of U Let us begin with anorienting observation

Proposition 421 An object c of CUχ is compact if and only if Oblv c is

Proof If c is compact then Oblv c is compact by the continuity of Avlowastψ If Oblv c is compactthen c is by the fully-faithfulness of Oblv

We will repeatedly use the following general observation to trace what duality does to compactobjects

Proposition 422 Suppose F C rarr D is a continuous functor between cocomplete dg-categoriesIf C and D are dualizable and F admits a continuous right adjoint G then the following diagramcommutes

CcopF

D

Dcop

D

CorcGor Dorc

Proof The continuity of G implies F preserves compactness If c isin Cc the asserted commutativityfollows from

〈Gorcorminus〉D 〈corGminus〉C HomC(cGminus) HomD(Fcminus) 〈(Fc)orminus〉D

Combining Propositions 415 422 we obtain


Corollary 423 For a strong representation C of U and a character χ U rarr Ga taking dualscommutes with Oblv ie we have a commutative diagram

CUχcop

D

Oblv Ccop

D

CorUminusχcOblv Corc

43 Whittaker invariants of dual representations of loop groups Having handled theprounipotent case we can now describe in the interactions between dualizable representations ofloop groups their Whittaker models and compact objects therein

431 Dualizability Write Dκ(LG) -mod for the (infin 2)-category of strong level κ representations ofLG If C in Dκ(LG) -mod is a dualizable object of DGCatcont then Cor naturally is a right module

for Dκ(LG) ie lies in Dminusκ(LG) -mod Writing Dκ(LG) -moddualizable for the full subcategory ofsuch objects we obtain a functor

D Dκ(LG) -moddualizable rarr Dminusκ(LG) -moddualizableop

Next let us explain why we did not consider twisted differential operators in the discussion for

prounipotent groups Let κprime be a level corresponding to a central extension LG of LG eg κKilling

Then on any ind-prounipotent subgroup U of LG the restricted central extension U uniquely splitsThen by ind-prounipotence it is split uniquely yielding a canonical identification for any κ

Dκ(U) -mod D(U) -mod (424)

By construction under these identifications dualizing commutes with restriction from Dplusmnκ(LG) toD(U)

In particular the above discussion applies to the subgroups LN and In n gt 1 showing up in theadolescent Whittaker construction As the Whittaker model involves multiple of these subgroups atonce let us note that for U prime sube U ind-prounipotent subgroups of LG by the uniqueness of splittings

of U prime U the following diagram commutes

Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res

Dκ(U prime) -mod(44)

D(U prime) -mod

We are now ready to state the interaction of duality and Whittaker models

Theorem 425 Let C be an object of Dκ(LG) -moddualizable with dual representation Cor Then fora nondegenerate character ψ N rarr Ga there is a canonical duality of Whittaker models

(CLNψ)or(Cor)LNminusψ

such that for any n gt 1 the following diagram commutes

CInψ otimes CorInminusψOblvotimesOblv[2n∆]

insotimes ins

Cotimes Cor

CLNψ otimes CorLNminusψ Vect

(426)

Remark 427 To remove the shift by [2n∆] in (426) the reader may prefer to write insteadCLNψorCorLNminusψ We choose to work with coinvariants in view of Wκ -mod

16 GURBIR DHILLON

Proof Let us recall the adolescent Whittaker presentations of the Whittaker model


CInψ CLNψ limlarrminusinm

CInψ

We will obtain the desired result from the following more general lemma

Lemma 428 Consider a diagram of categories Ci i gt 1 in DGCatcont

C1F21minusminusrarr C2

F32minusminusrarr C3F43minusminusrarr middot middot middot

Suppose that each Ci is dualizable and that Fi+1i are fully faithful and admit continuous rightadjoints Gii+1 Then there is a canonical perfect pairing

( limminusrarrFnm

Cn)or limminusrarrGormn

Corn

such that that for each n gt 1 the following diagram commutes

Cn otimes Corn

ampamp

insnotimes insn

limminusrarrCn otimes limminusrarrCorn Vect

(429)

where insn is the tautological map into the colimit

Proof This follows from [33] Lemma 222 where to obtain the diagram (429) one should noticethe simplification of Lemma 136 of loc cit in the case where the Fnm are fully faithful

To apply the lemma we recall that inmlowast was the composition



The latter relative averaging is by definition the right adjoint to Oblv CImψ rarr CImcapInψWe claim

that it may be calculated as the composition CIncapImψOblvminusminusminusrarr C

Avlowastψminusminusminusminusrarr CImψ Indeed we note thatthe fully-faithfulness of the forgetful functors CInψ rarr CCImcapInψ rarr C imply the fully-faithfulness

of CInψOblvminusminusminusrarr CIncapImψ from which we calculate for cnm isin CIncapImψ cm isin CImψ

HomCIncapImψ(Oblv cm cnm) HomC(Oblv cmOblv cnm) HomCImψ(cmAvlowastψ Oblv cnm)

By a similar argument for inm we have rewritten the functors as

inmlowast CInψOblvminusminusminusrarr C

Avlowastψminusminusminusminusrarr CImψ inm CorImminusψOblvminusminusminusrarr Cor

Avlowastminusψminusminusminusminusrarr CorInminusψ

By Proposition 49 it follows that inmlowast and inm are dual functors

We are ready to apply the lemma The Fnm are inmlowast the Gmn are imn[2(mminusn)∆] and henceby the preceding paragraph the Gormn are inmlowast[2(mminus n)∆] inm To conclude we use that

limminusrarrinmlowast

CorInminusψ limminusrarrinmlowast[2(mminusn)∆]

CorInminusψ

where we take the colimit of the functors id[2n∆] CorInminusψ rarr CorInminusψ


432 Dualizability and compact objects Whittaker models We first explain the relation betweencompact objects of the adolescent and full Whittaker models by proving the following generalproposition which roughly says taking compact objects commutes with taking unions in DGCatcont

Proposition 430 Let I be a filtered (infin 1)-category and Ci i isin I an I system of objects inDGCatcont Suppose for each 1-morphism α i rarr j in I the corresponding functor Fα Ci rarr Cjis fully faithful and admits a continuous right adjoint Gα Then for each i0 isin I the tautologicalfunctor Ci0 rarr limminusrarriisinI Ci preserves compactness and this induces an equivalence

limminusrarriisinI

Ccisimminusrarr (limminusrarr

iisinICi)

c (431)

Remark 432 To be clear in the left hand side of Equation (431) the colimit is taken in DGCatie is a colimit of non-cocomplete dg-categories and on the right hand side one is taking compactobjects from a colimit in DGCatcont

Proof For fixed ι isin I we claim the insertion insι Cι rarr limminusrarrCj is fully faithful and admits acontinuous right adjoint To see this we will show that under the isomorphism

limminusrarrFα

Ci limlarrminusGα

Ci

the desired right adjoint is given simply by the evaluation evι limlarrminusCi rarr Cι Indeed for an objectof limlarrminusCi which concretely is realized as a homotopy coherent system limlarrminus ci of objects of Ci i isin Iand an object c isin Cι we have

Hom(insι c limlarrminus ci) limlarrminusHom(evi insι c ci)

By filteredness of I we may without loss of generality run the above homotopy limit of complexesover indices i admitting a map β ι rarr i Moreover by the fully-faithfulness of the Fα for such iand β we have evi insι Fβc and hence

limlarrminusHom(evi insι c ci) limlarrminusHom(Fβc ci) limlarrminusHom(cGβci) Hom(c evι limlarrminus ci)Applying the above discussion of pairs i and β to ι and id ιrarr ι shows that evι is fully faithful

Having shown that insι admits a continuous right adjoint it follows that it preserves compactnessand hence we obtain the map (431) Recall that filtered colimits in DGCat can be computedvery naively with every object inserted from a step in the colimit and morphisms the colimit ofstepwise morphisms cf [56] By the fully-faithfulness of the Fα and insι it follows that (431)is fully-faithful To see that it is essentially surjective on homotopy categories observe that thecomposites insι evι ι isin I assemble into a colimit as one varies ι in a manner compatible with thecounits insι evι rarr id Moreover the resulting map limminusrarrιisinI insι evι rarr id is a natural isomorphism

Applying this to a compact object c of limminusrarrCi by compactness the inverse map c rarr insι evι cfactors through insi evi c for some i isin I Thus c is a direct summand of insi evi c and by thefully-faithfulness of insi moreover of the form insi c Again by the fully-faithfulness of insi it followsthat c is compact as desired

With these preparations we can state how duality acts on compact objects in Whittaker models

Theorem 433 Let C Cor ψ and (CLNψ)or(Cor)LNminusψ be as in Theorem 425 On compactobjects Theorem 425 combined with Propositions 416 430 yield


CInψcop CcopLNψC

orcLNminusψ limminusrarr

inmlowast

CorInminusψc

If we write Dn CInψcop rarr CorInminusψc for the equivalence induced by Proposition 415 then thecomposite equivalence

Dinfin limminusrarrinmlowast

CInψcop limminusrarrinmlowast

CorInminusψc

18 GURBIR DHILLON

is given by the colimit of [minus2n∆] Dn CInψcop CorInminusψc

Proof This follows from the construction of the pairings in Theorem 425 Proposition 416 andProposition 430 Nonetheless let us provide some detail for the convenience of the reader Letus write n〈minusminus〉 for the pairing between CInψ and CorInminusψ and infin〈minusminus〉 for the pairing betweenCLNψ and CorLNminusψ Then by construction for ξn isin CorInminusψ and cn isin CInψ we have

infin〈ins ξn ins cn〉 n〈ξn cn〉[2n∆]

To get the correct shifts for cn a compact object of CInψ and dn an arbitrary object of CInψ wecompute

infin〈(ins cn)or ins dn〉 Hom(ins cn ins dn) Hom(cn dn)

n〈corn dn〉 n〈corn [minus2n∆] dn〉[2n∆] infin〈ins(corn [minus2n∆]) dn〉

5 Categorical FeiginndashFuchs duality and semi-infinite cohomology for W-algebras

This section is structured as follows We first prove that semi-infinite cohomology gives a dualityfor Tate Lie algebras We then reverse this logic for W ie we apply the above duality in the caseof KacndashMoody and pass to Whittaker models to construct semi-infinite cohomology for W Wethen sketch an example calculation of this semi-infinite cohomology anticipated by I Frenkel andStyrkas

51 FeiginndashFuchs duality for Tate Lie algebras The goal of this subsection is Theorem 56which says that semi-infinite cohomology is a perfect pairing between Tate Lie algebra representa-tions at complementary levels For KacndashMoody this is a theorem of GaitsgoryndashArkhipov [4] Theproof of loc cit uses explicit kernel bimodules As we explain all that one needs is the Shapirolemma

511 Reminders on Tate Lie algebras Some references for Tate Lie algebras are Sections 2773817 of [5] and Appendix D of [52]

Let L be a Tate Lie algebra and write L -modhearts for the abelian category of smooth L repre-sentations By a central extension Lk of L we mean a Tate Lie algebra Lk fitted into an exactsequence

0rarr C1rarr Lk rarr Lrarr 0

where 1 isin Lk is central For a central extension Lk write Lk -modhearts for the full subcategory of itsabelian category of discrete modules on which 1 acts as the identity For any Tate Lie algebra Lone has its canonical Tate extension LTate For any compact open subalgebra L0 sub L one has acanonical section s0 L0 rarr LTate these need not be compatible under inclusions of compact opensubalgebras

For any compact open subalgebra L0 sub L one has the functor of semi-infinite cohomology

Cinfin2

+lowast(LminusTate L0minus) LminusTate -mod+ rarr Vect

For a concrete presentation ostensibly written in the setting of a Z-graded Lie algebra see Section2 of [19] Let us remind how this mildly depends on L0 Recall that for any two compact opensubspaces C0 C1 of a Tate vector space V one can form their relative determinant det(C0 C1) Ifwe choose a compact open subspace C2 containing C0 and C1 then there is a canonical isomorphism

det(C0 C1) det(C2C0)otimes det(C2C1)or

For two compact open subalgebras L0 L1 one has isomorphisms

Cinfin2

+lowast(LminusTate L0minus) Cinfin2

+lowast(LminusTate L1minus)otimes det(L0 L1)

where det(L0 L1) is graded by viewing L as a graded vector space of degree -1


The canonical section s0 L0 rarr LTate induces a section L0 rarr LminusTate hence one can inducerepresentations of L0 to LminusTate We will need the following standard

Proposition 51 (Shapiro lemma) For any M isin L0 -modhearts there is a canonical equivalence

Cinfin2

+lowast(LminusTate L0 indLminusTate

L0M) Clowast(L0M) (52)

Proof Since we are unaware of a reference containing a proof we sketch one Write Llowast for thecontinuous dual of L and Cl(L[1]oplus Llowast[minus1]) for the graded topological Clifford algebra generatedby L[1]oplusLlowast[minus1] with its tautological pairing This has a unique up to tensoring by a graded linesimple discrete graded module For a choice of open Lagrangian U sub L[1]oplusLlowast[minus1] one can realize

this module as indCl(L[1]oplusLlowast[minus1)]SymU C where C is the unique simple representation of the topological

exterior algebra SymU Let us write Sp for its realization associated to the Lagrangian L0 oplus Lperp0

Writing SpLperp0 for the subspace killed by Lperp0 we have

SpLperp0 = Sym(Llowast[minus1])C = Sym(LlowastLperp0 [minus1])C = Sym(Llowast0[minus1])C (53)

Recall the underlying vector space of Cinfin2

+lowast(LminusTate L0minus) is given by Sp otimes minus Equation (53)

identifies SpLperp0 otimesM with Clowast(L0M) and this is compatible with differentials

It remains to show this inclusion is a quasi-isomorphism To do so filter indM = indLminusTate

L0M

byF i indM = (F iU(LminusTate))M i gt 0

where U(LminusTate) denotes the universal enveloping algebra and F iU(LminusTate) is the ith step in itsPBW filtration Writing Spj j isin Z for the jth graded component of Sp filter Spotimes indM by

F iSpotimes indM =oplusminusj+k=i

Spj otimes F k indM

This is a filtration by subcomplexes In the associated spectral sequence for its cohomology onE1 one encounters a family of Koszul complexes for SymLL0 and the E2 page has the desiredform

512 Renormalized derived categories Recall that L is a topological Lie algebra with central ex-tension Lk Since the projection Lk rarr L is open one has

Lemma 54 Write CL for the category whose objects are compact open subalgebras C of L andwhose morphisms are inclusions Write CLk for the category whose objects are compact opensubalgebras Ck of Lk containing 1 and whose morphisms are inclusions Then pullback defines anequivalence CL CLk

Write Lk -modn (n for naive) for the usual unbounded derived category of Lk -modhearts Within

it consider the objects of the form indLkCk V where Ck isin CLk and V isin Ck -modhearts is a finite

dimensional module Consider the pretriangulated envelope C of these objects within Lk -modn6

and set Lk -mod = Ind(C) We have a tautological functor

Ψ Ind(C)rarr Lk -mod

Proposition 55 Ψ admits a (typically discontinuous) fully faithful right adjoint

Ψ Lk -mod Lk -modn Φ

Moreover there is a unique t-structure on Lk -mod with Lk -modgt0 = ΦLk -modngt0 In particularΦ and Ψ define inverse equivalences between the bounded below subcategories

6We could equivalently work with V isin Ck -modn with finitely many nonzero cohomology groups each smooth and

finitely generated or just indLkCk

C the trivial representations

20 GURBIR DHILLON

Proof Let us explain the fully faithfulness of Φ Since Lk -modhearts is a Grothendieck abelian cat-egory it follows that Lk -modn has enough homotopy injective complexes cf [60] WritingOblv Lk -modn rarr Vect for the forgetful functor and letting Ck run over the objects of CLk cf Lemma 54 it follows that

limminusrarrHom(indLkCk Cminus) Oblv(minus)

The remainder of the argument for fully faithfulness and those for the t-structures are now identicalto those in Section 22 of [25]

513 Semi-infinite cohomology and duality For a central extension Lk tensor product of repre-sentations gives a map

Lk -modotimesLminuskminusTate rarr LminusTate -mod

Namely one defines the pairing on our compact generators to be tensor product and then indextends Similarly recalling that L0 denotes a compact open subalgebra of L one obtains a map

Cinfin2

+lowast(LminusTate L0minus) LminusTate -modrarr Vect

We are now ready to prove the main result of this section

Theorem 56 The composite pairing

Cinfin2

+lowast(LminusTate L0minusotimesminus) Lk -modotimesLminuskminusTate rarr Vect (57)

is perfect ie is a duality datum in DGCatcont

Proof Let us write C = Lk -mod D = LminuskminusTate -mod and 〈minusminus〉 for the pairing (57) Sinceboth are compactly generated they are dualizable with duals and pairings

〈minusminus〉C C otimes Cor rarr Vect 〈minusminus〉D D otimesDor rarr Vect

The pairing (57) yields maps φ C rarr Dor γ Dor rarr C satisfying

〈minus γminus〉C 〈minusminus〉〈φminusminus〉DWe first observe that these preserve compactness Indeed fix a compact open K sub L pick a finitedimensional V isin Kk -modhearts and calculate

〈φ indLkKk Vminus〉D Cinfin2

+lowast(LminusTate L0 indLkKk(V )otimesminus) Cinfin2


KminusTate(V otimesminus))

Cinfin2

+lowast(LminusTateK indLminusTate

KminusTate(V otimesminus))otimes det(L0K) Clowast(KV otimesminus)otimes det(L0K)

HomKminuskminusTate -mod(V or otimes det(KL0)minus) HomLminuskminusTate -mod(indLminuskminusTate

KminuskminusTate(V or otimes det(KL0))minus)

Note that in the above calculation if minus is compact we are using the lsquonaiversquo definitions of the abovefunctors Since a general object minus of the renormalized derived category is a colimit of compactones the same calculation applies due to the continuity of renormalized functors appearing

Ie writing D for the equivalence D Dcop Dorc cf Proposition 416 we have produced anisomorphism

φ indLkKk V D indLminuskminusTate

KminuskminusTate(V or otimes det(KL0)) (58)

Interchanging the roles of C and D for any finite dimensional W isin KminuskminusTate -modhearts we obtain anisomorphism

γ indLminuskminusTate

KminuskminusTateW D indLkKk(Wor otimes det(KL0)) (59)

Having shown that φ preserves compactness we may apply the following general lemma

Lemma 510 Suppose CD are dualizable objects of DGCatcont equipped with a pairing C otimesD rarrVect Let φ γ be as above If φ preserves compactness and C is compactly generated then γ admitsa left adjoint γL


Proof Since φ preserves compactness we obtain a map φ Cc rarr Dorc Dcop and hence itsopposite DφD Ccop rarr Dc We will show that its ind-extension DφD Cor rarr D provides thedesired left adjoint If we pick an object ξ isin Corc this follows from

HomD(DφDξminus) 〈φDξminus〉D 〈Dξminus〉〈Dξ γminus〉C HomCor(ξ γ(minus))

To prove the theorem by the symmetry between C and D it suffices to show that γ is anequivalence Since γ is continuous preserves compactness and its essential image contains a set ofcompact generators of Cor it remains to show that γ is fully faithful on Dc To do so we will arguethat the natural transformation γLγ rarr idDc is an equivalence Let us introduce some notation

Set c = indLkKk V and c = Dc Set d = indLminuskminusTate

KminuskminusTate(V or otimes det(KL0)) Consider the diagram

Hom(γLc d)sim

(58)

〈c d〉 simHom(c γd)

(59)

Hom(γLc γLc) Hom(γd γd)

To see that γLγd rarr d is an isomorphism it remains to observe that the images of idγLc and idγdcoincide in 〈c d〉 Explicitly if we write

〈c d〉 Cinfin2

+lowast(LminusTateK indLkKk V otimes indLminuskminusTate

KminuskminusTateV or)

then both correspond to the image of idV under the composition

End(V ) V otimes V or rarr Clowast(KV otimes V or)rarr 〈c d〉

The following equivariance property of the pairing will be important to us It is known to expertsfor KacndashMoody Though we do not know of a published proof it will appear in forthcoming workof Raskin [55] see also [54]

Theorem 511 Suppose L is the Tate Lie algebra of an group ind-scheme H and suppose thatH contains a compact open subgroup K such that HK is ind-proper Then for any compact opensubalgebra L0 and central extension Lk of L the pairing

Cinfin2

+lowast(LminusTate L0minusotimesminus) Lk -modotimesLminuskminusTate -modrarr Vect

carries a canonical H equivariant structure

52 FeiginndashFuchs duality for W We would like to identify representations of W at complemen-tary levels κc plusmn κ as dual categories Let us do so now

Theorem 512 For any κ there is a canonical duality in DGCatcont

Wκ -modor Wminusκ+2κc -mod

Proof As in the previous subsection the functor of semi-infinite cohomology gives an LG equivari-ant perfect pairing

Cinfin2

+lowast(g2κc L+gminusotimesminus) gκ -modotimesgminusκ+2κc rarr Vect

Let us write ψ Lnrarr C for the character of Ln induced by the group homomorphism of Equation(42) and Cψ for the corresponding one dimensional representation For the time being let us

write Wψκ = H

infin2 (Ln L+nVκ otimes Cψ) As proved by Raskin [53] the functor

Cinfin2

+lowast(Ln L+nminusotimes Cψ) gκ -modrarrWψκ -mod

22 GURBIR DHILLON

induces an lsquoaffine Skryabinrsquo isomorphism

gκ -modLNψ Wψκ -mod (513)

Combining these with Theorem 425 we obtain

(Wψκ -mod)or (gκ -modLNψ)or gminusκ+2κc -modLNminusψ W

minusψminusκ+2κc

-mod (514)

It remains to explain why the difference between plusmnψ is inessential Namely writing H for theadjoint torus corresponding to the finite Cartan h sub g its conjugation action on N induces asimply transitive action on the nondegenerate ψ This gives an simply transitive action on thecorresponding Whittaker coinvariants

Lemma 515 For a character ψ of N and h isin H(C) consider the conjugated character hψ =ψ Adhminus1 Write δh isin D(H)hearts for the delta D-module supported on h Then for any category C

with a strong action of H n LN convolution with δh yields isomorphisms

δh minus CLNψ CLNhψ δh minus CLNψ CLNhψ (516)

Proof Let U be a quasicompact prounipotent subgroup of LN stable under the action of H andby abuse of notation continue to write ψ hψ for the restriction of these characters to U We firstshow the equivalence

δh minus CUψ rarr CUhψ (517)

To see this note that for ψ U rarr Ga by smoothness ψlowast Dlowast(U) rarr D(Ga) admits a left adjointψlowast Write eminusψ for the lsquotwisted constant sheafrsquo ψlowasteminusz[minus2] cf Equation (413) Then by the fully-faithfulness of Oblv CUψ rarr C CUψ may be recovered as the full subcategory of C consistingof the essential image of eminusψ minus Since δh δhminus1 = δe it is equivalently the essential image ofeminusψ δhminus1 minus Writing Adh LN rarr LN for conjugation by h we finish by computing

δh eminusψ δhminus1 Adhlowast e

minusψ (ψ Adhminus1)lowasteminusz[minus2] eminushψ (518)

Write LN as a union of compact open subgroups Uj j gt 1 stable under the action of H Recallthat

CLNψ limlarrminusOblv

CUj ψ CLNψ limminusrarrAvψlowast

CUj ψ

Thus to prove the lemma it suffices to verify the commutativity of the following diagrams for anyj gt i gt 1

CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus

CUj hψOblv CUihψ CUihψ

Avhψlowast CUj hψ

(519)

For the left hand diagram this is tautological and for the right hand diagram this follows fromEquation (518) applied to eminusψ on Uj Moreover a simple refinement argument using Equation(519) shows the resulting isomorphisms are independent of the choice of subgroups Uj j gt 1

Applying the lemma to KacndashMoody representations we obtain canonical isomorphisms for anynondegenerate characters ψψprime of N and level κprime

Wψκprime -mod gκprime -modLNψ gκprime -modLNψprime W

ψprime

κprime -mod

Concatenating this with Equation (514) yields the claimed duality

We finish this subsection by checking some compatibilites satisfied by the isomorphisms between

Wψκ -mod for varying ψ


Proposition 520 Let C be a Dκ(LG) module Then the isomorphisms of Lemma 515 are com-patible with the isomorphism between Whittaker invariants and coinvariants That is the followingdiagram commutes

CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ

Proof Let U be a prounipotent subgroup of LG stable under conjugation by H and consider acharacter χ of U Then the analog of Equation (517) remains true with a nearly identical proofSimilarly if U prime is another prounipotent subgroup of LG containing U and χ is the restriction of acharacter χprime of U prime then the analog of Equation (519) remains true

Briefly since every functor appearing in the construction of (44) is built from (co)limits ofcomposites of averaging and forgetful functors the required compatibility follows from (519)

More carefully recall that the functors inmlowast inm and inm cf Subsection 41 are compositesof averaging and forgetful functors Applying δh minus termwise to the appearing colimits and limitsyields by (519)

δh minus limminusrarrinmlowast

CInψ limminusrarrinmlowast

CInhψ limminusrarrinm

CInψ limminusrarrinm

CInhψ limlarrminusinm

CInψ limlarrminusinm

CInhψ

Writing I+n for the subgroup In cap LN we may use the I+

n as the subgroups Un n gt 1 appearingin Lemma 515 Concatenating the steps in the adolescent Whittaker construction we will showeach square in the following diagram commutes

limminusrarrAvψlowast

CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo

limminusrarrAvhψlowast

CI+n hψ limminusrarr

inmlowast

CInhψOblvoo

[minus] limminusrarrinm

CInhψsim

limlarrminusinm

CInhψ limlarrminusOblv

CI+n hψ

Avhψlowastoo

We claim that the squares (1) (2) and (4) commute before passing to (co)limits For the squares(1) and (4) this follows from Equation (519) For the square (2) this is tautological since bothhorizontal arrows are the same cohomological shift

For (3) it suffices to verify for any k gt j gt 1 the commutativity of the outer square of

CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus

CIj hψins limminusrarr

inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ

By the fully-faithfulness of the inm the horizontal composites are ijk whence the claim againfollows from Equation (519)

As a second compatibility note that the H integrability of Vκ yields an isomorphism of Lnmodules Vκ otimes Cψ Vκ otimes Chψ for any h isin H(C) Taking semi-infinite cohomology we obtain

24 GURBIR DHILLON

a canonical isomorphism Wψκ W

hψκ This tautologically gives an identification of their bounded

below derived categories of representations

Wψκ -mod+ Whψ

κ -mod+ (521)

This exchanges the compact generators for the renormalized unbounded categories yielding

Wψκ -mod Whψ

κ -mod (522)

The second compatibility we would like to record is between this isomorphism and affine Skyrabin

Proposition 523 The following diagram commutes

gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod

Proof This follows from the fact that δh minus on gκ -mod is restriction of representations alongAdhminus1

53 Semi-infinite cohomology for the W-algebra Recall that in the previous subsection wesaw a duality between Wκ -mod and Wminusκ+2κc -mod

Definition 524 The semi-infinite cohomology functor

Cinfin2

+lowast(minusotimesminus) Wκ -modotimesWminusκ+2κc rarr Vect (525)

is the pairing induced by Theorem 512

In the next remark we provide some orienting discussion

Remark 526 (1) Recall hor = minusκcκb

denotes the dual Coxeter number of g Recall that the

central charge of Wκ κ = kκb is given by

c(κ) = dim gk

k + horminus dim g + rk g + 24〈ρ ρ〉 minus 12(k + hor)κb(ρ ρ)

Since minusκ+2κc = (minuskminus2hor)κb it follows that the central charges for complementary levelsare again complementary

c(κ) + c(minusκ+ 2κc) = 2 rk g + 48〈ρ ρ〉Let us write cminusTate = 2 rk g + 48〈ρ ρ〉 for their common sum

(2) Let g be simply laced In this case we may equivalently write

c(κ) = rk gminus rk g(hor)(hor + 1)(k + hor minus 1)2

k + hor

If we consider the resulting map

P1 minushorinfin rarr P1 infin k 7rarr c(kκb)

the W-algebras corresponding to levels with the same central charge are identified by theFeiginndashFrenkel isomorphisms

W(minushor+ε)κb W(minushor+εminus1)κb ε isin CThus we may unambiguously write Wc for the W-algebra associated to g of central chargec and may rewrite (525) as

Wc -modotimesWcminusTateminusc -modrarr Vect c isin CThis latter parametrization was used in the known cases of semi-infinite cohomology namelyfor g = sl2 sl3 with cminusTate = 26 100 respectively


(3) Our functor (525) takes the shape of the so-called non-critical BRST reduction anticipatedin the conformal field theory literature For Wκ with c(κ) = cminusTate

7 it was anticipatedthere should be a critical reduction functor

Cinfin2

+lowast(minus) Wκ -modrarr Vect

This can be done by a similar method as we will explain in a subsequent publication

We bring two questions to the attention of the reader Firstly as suggested by Remark 526(2)there should be a compatibility between semi-infinite cohomology for W-algebras and FeiginndashFrenkelduality Second it would be good to establish the compatibility between the usual semi-infinitecohomology for Virasoro and one constructed in this subsection

For both one can show as proof of concept that the dualities coming from the two different con-structions of semi-infinite cohomology send the standard compact generators to the same objectsup to isomorphism

54 An example chiral differential operators In this subsection g = sl2 and κ is irrationalie κ isin Cκb Qκb We now address a problem raised by I Frenkel and Styrkas in Remark 7 of [27]ie to give a direct explanation of why the semi-infinite cohomology of the CDO for Virκ matchedthe semi-infinite cohomology of the CDO for gκ As anticipated in loc cit we show this may bedone via DrinfeldndashSokolov reduction

Since we will return to this question and others from loc cit in more generality elsewherewe only provide a sketch here Let P+ Zgt0 index the dominant integral weights of g and forλ isin P+ write Vκλ for the corresponding Weyl module of gκ As a gκoplus gminusκ+2κc module the Chiraldifferential operators for gκ decompose as

CDO oplusλisinP+

Vκλ otimes Vminusκ+2κcminuswλ

Using KacndashMoody duality we may calculate its semi-infinite cohomology as

Cinfin2

+lowast(g2κc L+gCDO)

oplusλisinP+

Homgκ -mod(VκλVκλ) Fun(GG)otimes Chg

where Chg denotes the Lie algebra cohomology Homg(CC) The chiral differential operators forVirκ are simply ΨΨ CDO where Ψ denotes DrinfeldndashSokolov reduction8 Accordingly

Ψ CDO oplusλisinP+

ΨVκλ otimesΨVminusκ+2κcλ

Using categorical FeiginndashFuchs duality we have

Cinfin2

+lowast(Vir26Ψ CDO) oplusλisinP+

HomVirκ -mod(ΨVκλΨVκλ)

It therefore suffices to argue that the natural map

Hom(VκλVκλ)rarr Hom(ΨVκλΨVκλ)

is an equivalence But since κ is generic in fact the corresponding block of monodromic category O

for gκ is sent isomorphically onto the corresponding block of monodromic category O for Virκ by ΨThis follows from the localization theorem alluded to at the end of Section 2 see also Subsection83

7In the reference [14] cminusTate is equivalently written as 2sums(6s

2 minus 6s + 1) where s runs over the conformal

dimensions of the standard generators for W ie the degrees of the fundamental invariants for g8We should mention that Ψ CDO depends on κ and not simply c(κ) More generally a basic fact of life is that the

two lsquoKazhdan-Lusztig categoriesrsquo in Wκ -mod coming g gL do not coincide Rather as explained to us by Creutzigthey form two lsquoaxesrsquo of a remarkable two parameter family of representations

26 GURBIR DHILLON

6 Jordan-Holder content and highest weight filtrations in Category O

Recall the definition of Category O from Subsection 36 We remind two basic properties ofCategory O for an affine Lie algebra First at a positive level κ many objects will not have finitelength Nonetheless one can make sense of the JordanndashHolder content of an object ie expressthe formal character of a module as a locally finite sum of simple characters Second any object ofO admits an ascending filtration with successive quotients highest weight modules In this sectionwe set up the analogous theory for W-algebras for W3 see [13]

61 Jordan-Holder content As a first observation note that since W is Zgt0 graded by L0 wehave the following coarse decomposition of O

Lemma 61 Let M be any object of O and write M =oplus

disinCMd for its L0 eigenspace decompo-sition as in Definition 38 For γ isin CZ consider the subspace

Mγ =oplusdisinγ+Z

Md

Then Mγ is a submodule of M and we have M =oplus

γisinCZMγ

Thus to define the Jordan-Holder content ofM it suffices to considerM = Mγ for some γ isin CZFix a lift γ isin C of γ For λ isinWfhlowast write 〈L0 λ〉 for its lowest energy ie the evaluation of λ onthe element of Zhu(Wκ) corresponding to the conformal vector ω

Lemma 62 An object M = Mγ admits a finite filtration

0 = M0 subM1 sub middot middot middot subMnminus1 subMn = M

such that each successive quotient MiMiminus1 is either (i) a simple module L(λi) with minus〈L0 λi〉 isinγ + Zgt0 or (ii) has minusL0 eigenvalues in γ + Zlt0

Proof By assumption D = dimoplus

nisinZgt0 Mγ+n ltinfin Take n maximal for which Mγ+n is nonzeroThen Zhu(W) acts on Mγ+n and we may take an eigenvector v with eigenvalue λ isin Wfhlowast Thisinduces a map M(λ) rarr v whose image we call M prime There is a short exact sequence 0 rarr N prime rarrM prime rarr L(λ) rarr 0 which we use to form a filtration N prime sub M prime sub M By induction on D we mayproduce filtrations of N prime and MM prime of the desired form The induced filtration of M satisfies theconditions of the lemma

We would like to say for λ isin Wfhlowast minus〈L0 λ〉 isin γ + Zgt0 that [M L(λ)] should be the numberof times in a filtration as above the successive quotient is isomorphic to L(λ)

To do so consider Oγ the full subcategory of O consisting of objects with minusL0 eigenvalues inγ+Z Within Oγ consider Oltγ the full subcategory of O consisting of objects with minusL0 eigenvaluesin γ + Zlt0 By construction we have

Lemma 63 Oltγ is a thick subcategory of Oγ ie is a full abelian subcategory closed undersubquotients and extensions

As with any thick subcategory of an abelian category one can form the quotient abelian categoryOγOltγ The objects are again those of Oγ but the morphisms between M and N are the directlimit over Hom(M prime NN prime) where M prime is a subobject of M with MM prime isin Oltγ and N prime isin Oltγ is asubobject of N

Proposition 64 In the quotient category OγOltγ every object is of finite length and the isomor-phism classes of irreducibles are given by the L(λ)minus〈L0 λ〉 isin γ + Zgt0

Proof It follows from Lemma 62 that every object is of finite length and any simple object is ofthe form L(λ)minus〈L0 λ〉 isin γ + Zgt0 From the explicit form of morphisms and using the simplicityof the L(λ) it is easy to see the L(λ)minus〈L0 λ〉 isin γ + Zgt0 remain mutually non-isomorphic


With this we can prove

Theorem 65 Fix M isin Oγ and L(λ) with minus〈L0 λ〉 isin γ+Z If we pick a γ with minus〈L0 λ〉 isin γ+Zgt0and a filtration of M as in Lemma 62 then the number [M L(λ)] of times MiMiminus1 is isomorphicto L(λ) is independent of the choice of γ and the filtration

Proof For fixed γ the independence from the choice of filtration follows from the fact that [M L(λ)] is really the Jordan-Holder content in the quotient category OγOltγ To compare the number[M L(λ)] obtained from γ and γminusn n isin Zgt0 one can refine a filtration for γ as in Lemma 62 toobtain one for γ minus n

For a general object M of O and L(λ) with minus〈L0 λ〉 isin γ+Z we define [M L(λ)] = [Mγ L(λ)]where Mγ was defined in Lemma 61 We now collect the basic properties of this Jordan-Holdercontent

Proposition 66 For a short exact sequence 0 rarr M prime rarr M rarr M primeprime rarr 0 and any λ isin Wfhlowast wehave [M L(λ)] = [M prime L(λ)] + [M primeprime L(λ)]

Proposition 67 An object M of O is zero if and only if [M L(λ)] = 0 for all λ isinWfhlowast

62 (Co-)Highest weight filtrations Recall the basic operation of picking a highest weightvector in an object of Category O which already appeared in Lemma 62 Iterating this we obtain

Proposition 68 Suppose M = Mγ for some γ isin CZ Then M admits an ascending filtration

0 = M0 rarrM1 rarrM2 rarr middot middot middot limminusrarri

Mi = M

such that each successive cokernel MiMiminus1 i gt 1 is a highest weight module

Proof Begin as in Lemma 62 to obtain M prime a highest weight submodule of M whose highest weightline lies in the maximal nonzero Md d isin γ+Z We set M1 = M prime and apply the same constructionto MM1 to produce M2 etc Since we use a maximal nonzero Md at each step and the weightspaces Md d isin C are all finite dimensional it is clear this construction satisfies the conditions ofthe proposition

By applying the duality D on Category O we obtain a dual form of Proposition 68

Proposition 69 Suppose M = Mγ for some γ isin CZ Then M admits a descending filtration

middot middot middot Aminus2 Aminus1 A0 = 0 limlarrminusi

Ai = M

such that each successive kernel ker(Ai rarr Ai+1) i 6 minus1 is a co-highest weight module

Remark 610 It may be psychologically helpful for the reader to note that in the the inverselimit of Proposition 69 for each Md d isin C the inverse limit limlarrminusi(Ai)d stabilizes after finitely manysteps

Remark 611 The results and proofs in this section should apply verbatim to any Zgt0 gradedvertex algebra which is finitely strongly generated cf [24]

7 The abstract linkage principle

In this section and its sequel we will canonically decompose Category O for Wκ as a direct sumover images of W orbits in Wfhlowast For nonintegral weights and levels the blocks should as usualrefine the above decomposition In this first section we give a general abstract decomposition of Ointo blocks For KacndashMoody algebras this is done in [18] [49] by similar means albeit with less useof co-highest weight filtrations In Section 8 we will then provide a Coxeter theoretic description

28 GURBIR DHILLON

We now introduce notation For micro ν isin Wfhlowast write micro ≺ ν if L(micro) is a subquotient of M(micro)Consider the finest partition

Wfhlowast =⊔α

Pα

such that if ν isin Pα and ν ≺ micro then micro isin Pα For fixed Pα write Oα for the full subcategory of Oconsisting of objects M all of whose simple subquotients have highest weights in Pα We will referto Pα as a linkage class and to Oα as the corresponding block Our goal is to prove the followingtheorem which gives the block decomposition of O

Theorem 71 Every object M of O canonically decomposes as a sumoplus

αMα where Mα lies inOα

We will obtain Theorem 71 from the following theorem which we turn to next

Theorem 72 For objects M isin Oα and A isin Oαprime α 6= αprime Ext1(MA) = 0

We now begin the proof of Theorem 72

Lemma 73 For objects M isin Oα and A isin Oαprime Hom(MA) = 0 In particular if an extension ofM by A splits it does so uniquely

Proof The first point follows from considering the JordanndashHolder content of the image of a mor-phism which would necessarily vanish cf Propositions 66 67 The second follows since the setof splittings is a torsor over Hom(MA)

Next we check the desired Ext vanishing for Verma and co-Verma modules

Lemma 74 For λ and ν isinWfhlowast λ 6= ν Ext1(M(λ) A(ν)) = 0

Proof Using the duality D on O we have a canonical isomorphism

Ext1(M(λ) A(ν)) Ext1(M(ν) A(λ))

Since q-characters are unchanged by duality by using the above isomorphism we may assumewithout loss of generality that minus〈L0 ν〉 isin minus〈L0 λ〉 + Zgt0 Suppose we have an extension 0 rarrA(ν)rarr E rarrM(λ)rarr 0 Writing d = minus〈L0 λ〉 ie the highest minusL0 eigenvalue of M(λ) we havean exact sequence

0rarr A(ν)d rarr Ed rarrM(λ)d rarr 0 (75)

By our assumption Ed+n = 0 for n isin Zgt0 hence Ed consists of singular vectors Viewing Equation(75) as a short exact sequence of Zhu(W) modules since λ 6= ν the sequence is split uniquely Bythe universal property of M(λ) the corresponding morphism M(λ)rarr E is again a splitting

We now deduce the desired statement by devissage as explained in the following lemmas

Lemma 76 For a highest weight module M isin Oα and A(ν) isin Oαprime Ext1(MA(ν)) = 0

Proof By assumption we have an exact sequence 0rarr K rarrM(λ)rarrM rarr 0 where M(λ) and Kare objects of Ol This gives an exact sequence

Hom(KA(ν))rarr Ext1(MA(ν)rarr Ext1(M(λ) A(ν))

where Hom(KA(ν)) and Ext1(M(λ) A(ν)) vanish by Lemmas 73 and 74 respectively

Lemma 77 For a highest weight module M isin Oα and a co-highest weight module A isin OαprimeExt1(MA) = 0

Proof Similarly to the proof of Lemma 76 we use an exact sequence

0rarr Ararr A(ν)rarr C rarr 0

and finish by the statements of Lemmas 73 and 76


We deduce

Lemma 78 If M isin Oα has a finite filtration with successive quotients highest weight modulesand A isin Oαprime is a co-highest weight module then Ext1(MA) = 0

Proposition 79 If M isin Oα is arbitrary and A isin Oαprime is a co-highest weight module thenExt1(MA) = 0

Proof Write M as a colimit limminusrarriMi as in Proposition 68 and consider an extension

0rarr Ararr E rarr limminusrarri

Mi rarr 0

We will split it on the right For fixed i consider the pullback

0rarr Ararr Ei rarrMi rarr 0

To split E it suffices to produce splittings si Mi rarr Ei which are compatible as we vary i But Mi

by definition admits a finite filtration with successive quotients highest weight modules whence byLemma 78 admits a splitting si If i 6 j then the pullback of sj produces a splitting of Ei Hencethe compatibility of the si follows from their uniqueness cf Corollary 73

Similarly to Lemma 78 we deduce

Lemma 710 If M isin Oα is arbitrary and A isin Oαprime has a finite filtration with successive quotientsco-highest weight modules Then Ext1(MA) = 0

Finally an argument dual to that of Proposition 79 finishes the devissage

Proof of Theorem 72 Write A as an inverse limit limlarrminusiAi as in Proposition 69 Given an extension

E isin Ext1(MA) one splits it on the left by giving compatible splittings of the pushouts

0rarr Ai rarr Ei rarrM rarr 0

as in the proof of Proposition 79

Proof of Theorem 71 Let M be an arbitrary object of O By Lemma 61 we may reduce to thecase M = Mγ γ isin CZ Write M as a colimit limminusrarri

Mi as in Proposition 68 For each i Mi has afinite filtration with successive quotients highest weight modules By Theorem 72 Mi accordinglyhas a finite direct sum decomposition of the desired form which we write as Mi =

oplusαMiα By

Lemma 73 the morphism Mi rarrMj decomposes as a sum of morphisms Miα rarrMjα Accordinglywe have

M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα

which has the desired form

Corollary 711 For M isin Oα and A isin Oαprime α 6= αprime we have

ExtiO(MA) = 0 i gt 0

Remark 712 Since we are working with a monodromic Category O it would be good to determinewhether the map OrarrW -modhearts is derived fully faithful

Remark 713 A similar argument applies mutatis mutandis to any Zgt0 graded vertex operatoralgebra which is finitely strongly generated

30 GURBIR DHILLON

8 The Coxeter theoretic linkage principle

Let us recall that for KacndashMoody Lie algebras one can prove an abstract linkage principle aswas done in the previous section This reduces the classification of blocks to a question about theJordan-Holder content of Verma modules The latter question in turn has a beautiful answer interms of the dot action of the Weyl group on hlowast and is the starting point for the Kazhdan-Lusztigcombinatorics

In this section we will perform the analogous classification for W Whereas for gκ blocks wereparametrized by the cosets of a Coxeter group modulo a parabolic subgroup for Wκ the blocks areparmetrized by double cosets of a Coxeter group modulo two parabolic subgroups Loosely the onesided quotient is what one had prior to DrinfeldndashSokolov reduction and the new quotient comesfrom the projection hlowast rarrWfhlowast

81 The linkage principle recollections for KacndashMoody Let κ be an arbitrary noncriticallevel We now review the classification of blocks for gκ

Let λ isin hlowastκ The block passing through λ will be controlled not always by the full affine Weylgroup W but rather a subgroup which we now recall To do so we will need some notationWrite Φor+ for the positive coroots of gκ Φorminus for the negative coroots of gκ and Φor = Φor+ t Φorminus forthe coroots Write Φorre for the real coroots and similarly Φorplusmnre for the positive and negative realcoroots The choice of λ affords the subset

Φorλ = α isin Φor 〈α λ+ ρ〉 isin ZIntersecting Φorλ with the real and positive or negative coroots yields

Φorλplusmn = Φorλ cap Φorplusmn Φorλre = Φorλ cap Φorre Φorλreplusmn = Φorλ cap Φorplusmnre

The reflections sα corresponding to every real coroot in Φorλ generate a subgroup Wλ of W theintegral Weyl group Wλ = 〈sα〉 α isin Φorλre With this we have the following well-known

Proposition 81 For λ isin horκ the block of Category O for gκ containing L(λ) has highest weightsWλ middot λ

We now recall the Coxeter theoretic parametrization of the block Within Φorλ one has the lsquosimpleintegral corootsrsquo

Πλ = α isin Φorλ+re sαΦorλ+ cap Φorλminus = minusαEquivalently Πλ consists of the elements of Φorλ+re which cannot be expressed as the sum of two

other elements of Φorλ+ If we index the simple integral roots by Iλ ie

αi i isin Iλ = Πλ

then it is known that Wλ is a Coxeter group with simple reflections si i isin IλThe orbit Wλ middotλ will pass through a unique dominant or antidominant weight depending on the

sign of κκ isin κc + Qgt0 Cminusκ = λ isin hlowastκ 〈λ+ ρ α〉 isin Zgt0 α isin Φor+reκ isin κc + Qgt0 C+

κ = λ isin hlowastκ 〈λ+ ρ α〉 isin Zlt0 α isin Φor+re (82)

Note that the above dichotomy while a standard practice in the literature is partially a conventionas nonintegral blocks may contain both a dominant and antidominant weight and one can equallywell parametrize κ isin Qκc by dominant weights

Observe also that we can rewrite (anti)dominance in terms of the integral Weyl group

Lemma 83 We have the equalities

Cminusκ = λ isin hlowastκ 〈λ+ ρ αi〉 isin Z60 i isin Iλ

C+κ = λ isin hlowastκ 〈λ+ ρ αi〉 isin Zgt0 i isin Iλ (84)


Proof Since αi i isin Iλ lie in Φorλ we have 〈λ+ ρ αi〉 isin Z It follows that the conditions defining theright hand side of (84) are a subset of those in (82) To see the reverse inclusion for α isin Φor+reΦorλ 〈λ + ρ α〉 isin Z and hence the condition of (82) is automatic for such α Moreover if we knowthe right hand side of (84) the fact that (82) holds for all α isin Φorλre+ follows from knowing that

Φorλ+ sub Zgt0αi i isin Iλ

For λ isin Cplusmnκ the stabilizer of λ in Wλ which we denote by W λ is generated by the simplereflections fixing λ

82 The linkage principle the W-algebra Having recalled the classification of blocks for gκwe now perform the analogous classification for Wκ

The main subtlety is that for λ in Wfhlowastκ there may be several Wf antidominant weights λ isin hlowastκwith image λ To say how many write fΦor for the finite coroots ie associated to g and consider

fΦorλ = α isin fΦor 〈λ+ ρ α〉 isin ZWe may again consider the subgroup Wfλ sub Wf generated by the reflections sα α isin fΦorλ Thenthere are WfWfλ many Wf antidominant weights in Wf middotλ as we develop in the next two lemmas

Lemma 85 Write W fλ for the stabilizer of λ in Wf Then W fλ is contained in Wfλ

Proof This is well known and can be proved by adapting the argument of Lemma 87

Lemma 86 For each coset yWfλ isin WfWfλ there is a unique Wf antidominant weight inyWfλ middot λ In particular under the identification of Wf middot λ WfW

fλ each fibre of the projection

onto WfWfλ contains a unique Wf antidominant weight

Proof It is straightforward that there is a unique Wf antidominant weight in Wfλ middotλ cf the proofof Lemma 83 To finish the first part of the lemma note that for any y isinWf Wfymiddotλ = yWfλy

minus1and hence Wfymiddotλ middot(y middotλ) = yWfλ middotλ The remainder of the Lemma now follows from Lemma 85

The previous two lemmas describe Wf antidominance within a single Wf orbit We next need tounderstand Wf antidominance within a single W orbit or an integral Weyl group Wλ orbit therein

Lemma 87 Let λ isin hlowastκ κ 6= κc Write W λ for the stabilizer of λ in W Then W λ is contained inWλ

Proof It suffices to see that W λ is generated by the affine reflections sα α isin Φorre it contains Wemoreover claim it suffices to check this for a single κ 6= κc Indeed the scaling action of Gm onhlowast centered at minusρ yields W equivariant isomorphisms between all noncritical hlowastκ

We will take κ isin κc+Rgt0κb Note that for such κ hlowastR the real affine span of the integral weightsat that level is stable under the level κ dot action of W on hlowast Let us denote this restricted actionby hlowastRκ Under the decomposition hlowastκ = hlowastR oplus ihlowastR W acts diagonally The first factor transformsas hlowastRκ and the second transforms as the level zero unshifted action of W on hlowastR

Let us write λ isin hlowastκ0= λre + iλim We may compute W λ by first computing the stabilizer of λre

and then taking the subgroup of it which fixes λim By our choice of κ we may replace λ by a Wtranslate so that λre lies in the positive alcove

C+ = ν isin hlowastR 〈ν + ρ αi〉 isin Rgt0 i isin IThe stabilizer of λre is then a parabolic subgroup WJ of W for a proper subset J sub I of the affinesimple roots

Because J was a proper subset of I we may further act by WJ to carry λim into

C+J = ν isin hlowastR 〈ν αj〉 isin Rgt0 j isin J

It follows that the stabilizer of λim in WJ is generated by the simple reflections sj j isin J it containsas desired

32 GURBIR DHILLON

Lemma 88 Fix λ isin hlowastκ κ 6= κc and ν isinWλ middot λ Then the intersection of Wf middot ν and Wλ middot λ is (i)acted on transitively by Wfλ and (ii) contains a unique Wf antidominant weight

Proof We start with (i) ie that

Wf middot ν capWλ middot λ = Wfλ middot ν

The inclusion sup is straightforward To show sub let us write ν = w middot λ for some w isin Wλ Supposethat we have yw middot λ = wprime middot λ for some y isin Wf w

prime isin Wλ Then we have wminus1yminus1wprime middot λ = λ ByLemma 87 wminus1yminus1wprime isinWλ As wwprime isinWλ it follows that y isinWλ as well Therefore it remainsto show that

Wλ capWf = Wfλ (89)

The inclusion sup is again clear For the inclusion sub it suffices to show that WλcapWf is generated bythe reflections sα α isin fΦor it contains We will show that Wλ capWf is in fact a parabolic subgroupof Wλ

To see this recall that the simple positive roots in Φorλ were those α isin Φorλ+re which could not be

written as αprime + αprimeprime for αprime αprimeprime isin Φorλ+ and similarly for fΦorλ Since fΦor is the root system of a Levi

of Φor it follows that α isin fΦorλ+ is simple in fΦorλ if and only if it is simple in Φorλ Thus Wfλ is aparabolic subgroup of Wλ In particular it may be characterized as those w isin Wλ which preserveΦorλ+ fΦorλ+ But any w isin Wλ capWf shares this property and hence Wλ capWf = Wfλ as desired

This completes the proof of (i)To prove (ii) it suffices to show that Wfλ = Wfν But recalling that Wλ = Wν we may use

Equation (89) to conclude

Wfλ = Wλ capWf = Wν capWf = Wfν

as desired

With these preparations we may prove the linkage principle

Theorem 810 Let κ be noncritical and λ isinWfhlowastκ Then

(1) For any lift λ isin hlowastκ of λ the linkage class of λ is given by the image of Wλ middot λ in Wfhlowastκ(2) Write W λ for the stabilizer of λ under the level κ dot action of Wλ on hlowastκ Then the highest

weights in the linkage class of λ are parametrized by WfλWλWλ

Proof We start with (1) Let ν isinWfhlowastκ be arbitrary and λprime lie in the image of Wλ middot λ It sufficesto show that if either ν ≺ λprime or ν λprime cf Section 7 for the notation then ν is also in the image ofWλ middot λ As the statement of the theorem is unchanged by replacing λ with λprime we may take λprime = λ

First consider the case ν ≺ λ ie [M(λ) L(ν)] gt 0 Before Drinfeld-Sokolov reduction itfollows that [M(λ) L(ν)] gt 0 for some lift ν of ν By the block decomposition for gκ ν lies inWλ middot λ as desired

Next consider the case λ ≺ ν Consider an arbitrary lift ν isin hlowastκ of ν Before Drinfeld-Sokolovreduction we deduce that [M(ν) L(η)] gt 0 for some Wf antidominant η lying in Wf middot λ If wewrite η = y middot λ y isinWf then the block for gκ passing through η is Wη middot η = yWλ middot λ In particularν lies in yWλ middot λ and hence ν lies in the image of Wλ middot λ This concludes the proof of (1)

Statement (2) follows from combining (1) and Lemma 88 Namely we may identify Wλ middot λ withWλW

λ and Lemma 88 identifies the Wf antidominant weights in Wλ middot λ with WfλWλW

λ

83 Some conjectures on the blocks of W Let λ be antidominant and consider WfλWλWλ

This exhibits the highest weights of its block as a double coset of a Coxeter group by two parabolicsubgroups We conjecture the block of O only depends on this To this end it likely admits adescription as appropriate monodromic parabolic-singular Soergel modules In particular blocksshould have graded lifts and Koszul duality


A related expectation is that one should have translation functors relating various blocks for Wwhich should realize some of the above equivalences This was mentioned as a folklore conjecture in[13] With some care one can construct translation functors from those for gκ via affine Skyrabinas we hope to explain elsewhere We were informed by Creutzig of an alternative approach totranslation functors which has been developed in forthcoming work by him and Arakawa

Next let λ be regular antidominant so that we are dealing with a one sided coset WfλWλ Inthis case our conjecture asserts this block is equivalent to a monodromic singular block of O forthe appropriate KacndashMoody algebra Moreover at least in this case we conjecture that DrinfeldndashSokolov reduction should identify with a translation functor To state this precisely suppose forsimplicitly that λ is integral and write Omonλ for the corresponding block of monodromic CategoryO for gκ and OWκλ for the corresponding block of O for Wκ Note that Omonλ may be identified withthe free monodromic Hecke algebra and hence carries an involution ι coming from inversion on thegroup Finally write T for a translation functor from Omonλ to a Wf singular block of monodromicCategory O which we denote by Omonminusρ Then we conjecture an equivalence OWκλ Omonminusρ fittinginto a commutative diagram

Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ

For nonintegral λ one should replace gκ by a Kac-Moody algebra corresponding to its integralWeyl group with a appropriately modified torus or consider the neutral block of the correspondingtwisted free monodromic Hecke algebra for gκ We suspect that several but not all cases of thisconjecture should follow from our forthcoming work on the tamely ramified FLE

Analogous statements for λ dominant should be true In particular there should exist appro-priately defined positive level Soergel modules Moreover all of these should have non-monodromicvariants given a good definition for non-monodromic Category O for W For Virasoro translationfunctors were pursued and Koszulity was studied in [12] [64]

Similar stories should hold for affine W-algebras associated to other nilpotent elements principalin a Levi Finally we were happy to learn that for the finite W-algebras a completely parallel storyhas been largely been worked out [15] [39] [47] [48] [63] We thank Ben Webster for bringing thisto our attention

Remark 811 For ease of notation we wrote this section over the complex numbers C Howeverthe same results hold with similar proofs for any algebraically closed field k of characteristic zeroNamely one may deduce the case of Q from that of C Lemmas 85 87 may then be proven for kby choosing a basis for k over Q in a similar to manner to how we deduced them for C from thestatements for R

9 Verma embeddings

In this section we prove de Vostndashvan Drielrsquos conjecture on homomorphisms between Vermamodules

91 Negative level In this subsection κ will be negative ie κ isin κc + Qgt0κb We will nowdetermine the homomorphisms between all Verma modules for Wκ

For Verma modules MM prime in distinct blocks Hom(MM prime) = 0 cf Lemma 73 Therefore wewill fix a λ isin Wfhlowastκ and study the Verma modules within its block Recall that if we pick a liftλ isin hlowastκ of λ the highest weights of the block through λ are parametrized by WfλWλW

λ For a

double coset w isinWfλWλWλ let us write Mw for the corresponding Verma module

Recall that Wλ is a Coxeter group with simple reflections si i isin Iλ In Lemma 88 it was shownthat Wfλ is a parabolic subgroup of Wλ By possibly replacing λ by a Wλ translate we may assume

34 GURBIR DHILLON

λ is antidominant In this case its stabilizer W λ is again a parabolic subgroup of Wλ cf the proofof Lemma 87

As a quotient of a Coxeter group by two parabolic subgroups WfλWλWλ inherits a partial

order from the Bruhat order on Wλ which we denote by 6 Explicitly each double coset admits aunique minimal length representative and one restricts the Bruhat order on Wλ along this sectionWith these preliminaries we may now state the main result of this subsection

Theorem 91 For elements v w isinWfλWλWλ Hom(MvMw) is at most one dimensional and

is nonzero if and only if v 6 w

911 Negative level (i) recollections for KacndashMoody To prove Theorem 91 we will need theanalogous story for KacndashMoody

Proposition 92 Fix an antidominant weight λ isin hlowastκ and for w isin WλWλ write Mw for the

Verma module M(w middot λ) Then

(1) The antidominant Verma Me is simple and [My Me] = 1 for any y isinWλWλ

(2) Hom(MyMw) is at most one dimensional and is nonzero if and only if y 6 w in theBruhat order on WλW

λ

Proof We first recall that Wλ is the Weyl group of a sum of KacndashMoody algebras of finite andaffine type To see this let I sub Iλ correspond to an indecomposable summand ie a connectedcomponent of the Dynkin diagram Write δ for the indecomposable positive imaginary root of gκThen the Cartan matrix of I is of infinite type if and only if δ lies in the real span of the αι ι isin Imoreover if δ lies in the real span of αi i isin I it in fact may be written as

sumi ciαi with all ci

positive cf Lemmas 22 and 23 of [45] Since 〈δ αi〉 = 0 i isin I it follows that I is of affine typeby Proposition 47 of [43]

By a result of Fiebig [22] the Proposition is reduced to the analogous assertion for a negativeintegral block for the KacndashMoody algebra of type Iλ Moreover it is straightforward to reduce theassertion to the analogous one for each I sub Iλ which is indecomposable To do so one may use thecorresponding tensor product factorization of Verma and simple highest weight modules

Therefore the Proposition is reduced to the case of an integral negative block for an affineLie algebra or an integral block of a finite dimensional simple Lie algebra As the latter is welldocumented we only discuss the former By using translation functors from a regular to a singularblock we may further reduce to the case of a regular integral block

It remains to verify the Proposition for a regular integral block at negative level for an affine Liealgebra l with Weyl group Wl For (1) that Me is simple goes back to KacndashKazhdan [42] That[My Me] = 1 for y isinWl is a consequence of KazhdanndashLusztig theory Namely [My Me] is givenby the inverse KazhdanndashLusztig polynomial Qey(1) To see that Qey(1) = 1 one may eg use theinterpretation of Qey(1) at positive level and compare with the WeylndashKac character formula

To see (2) the analogous claim at positive level is proved in Proposition 255 of [44] Todeduce the claim at negative level one may apply KacndashMoody duality by the argument of Lemma98 Alternatively the characterization of when there are nonzero intertwiners follows from KacndashKazhdan and their uniqueness follows from (1) cf the proof of Theorem 91

912 Negative level (ii) embeddings for W First let us show

Proposition 93 Let κ be arbitrary A nonzero morphism between Verma modules for Wκ isinjective

Proof Let M(λ) be a Verma module Recall the Z-graded current algebra U(W) cf [2] The usualfiltration on W induces a filtration on the associated Lie algebra of Fourier modes and a filtrationF iU(W) i gt 0 Write Wi 1 6 i 6 rk g for the usual generators of Wκ Write Wi(n) n isin Z for the


Fourier mode of Wi of degree n and W i(n) for its symbol in grU(W) Then grU(W) contains thepolynomial algebra

P = C[W i(n)] 1 6 i 6 rk g n gt 0

M(λ) is cyclically generated as a U(W) by its highest weight vector vλ Under the induced filtrationF iM(λ) i gt 0 of M(λ) grM(λ) is exhibited as a free P module with generator the symbol of vλcf Section 51 of loc cit

Given a nonzero map φ M(λ) rarr M(ν) write FnM(ν) for the minimal filtered piece of M(ν)containing vλ It follows that φ carries F iM(λ) into F i+nM(ν) It suffices to check injectivity onthe associated graded which is clear since this is a nonzero map of rank one P modules

Having gathered the necessary ingredients we will determine the homomorphisms between Vermamodules for W at negative level

Proof of Theorem 91 We first claim that Me is simple and [Mv Me] = 1 for all v in WfλWλWλ

To see this one may apply the DrinfeldndashSokolov reduction DSminus cf Theorem 311 to Proposition92(1)

We next claim that Hom(MeMv) is one dimensional Indeed that it is at most one dimensionalfollows from the facts that [Mv Me] = 1 and that Mv is of finite length That it is one dimensionalfollows by applying DSminus to a nonzero morphism in gκ which exists by Proposition 92(2) That itremains nonzero follows from the exactness of DSminus

We next claim show Hom(MvMw) is at most one dimensional for any v w isin WfλWλWλ

To see this suppose one has f g isin Hom(MvMw) By the preceding paragraph we may multiplyf by a scalar and assume that f minus g vanishes on the copy of Me embedded in Mv As a nonzerohomomorphism between Verma modules is an embedding cf Proposition 93 it follows that f = gas desired

We finally claim that Hom(MvMw) is nonzero if and only if v 6 w Let us write s(v) s(w) isinWλ

for their minimal length coset representatives Suppose that v 6 w ie that s(v) 6 s(w) ByProposition 92 there is an embedding Ms(v) rarrMs(w) Applying DSminus one obtains an embeddingMv rarr Mw For the reverse implication suppose one has an embedding Mv rarr Mw In particularone has [Mw Lv] gt 0 Before applying DSminus one has [Ms(w) Lsprime(v)] gt 0 where sprime(v) is theunique Wf antidominant preimage of v in Wλ middot λ cf Lemma 88 It follows that sprime(v) 6 s(w) Ass(v) 6 sprime(v) by standard Coxeter combinatorics we have v 6 w as desired

Remark 94 Although it is not necessary for the proof one can show that

s sprime WfλWλWλ rarrWλ

coincide ie s = sprime

92 Positive level We now explain how to deduce the classification of Verma embeddings atpositive level from that at negative level via FeiginndashFuchs duality As a preparatory observationnote that under the linear action of Wf on hlowast negation sends Wf orbits into Wf orbits Similarlyunder the dot action of Wf on hlowast shifted negation ν rarr minusνminus2ρ descends to an involution of Wfhlowast

Theorem 95 For κ noncritical and λ isin Wfhlowastκ write Mκ(λ) for the Verma module for Wκ ofhighest weight λ Then under Feigin-Fuchs duality we have

DMκ(λ) Mminusκ+2κc(minusλminus 2ρ)[dimN ]

In particular writing Vermaheartsκ for the full subcategory of Wκ -modhearts consisting of Verma modules

we have a contravariant equivalence Vermaheartsopκ Vermaheartsminusκ+2κc

Proof We first check that Mκ(λ) is a compact object of Wκ -mod Write I Iminus for the Iwahorisubgroups of L+G corresponding to the preimages of B and Bminus under the evaluation map L+Grarr

36 GURBIR DHILLON

G respectively Recall we wrote I prime1 for the prounipotent radical of I and I1 = Adtminusρ Iprime1 Similarly

let us write Iprimeminus1 for the prounipotent radical of Iminus and Iminus1 = Adtminusρ I

primeminus1

Following the conventions of [53] we view the minus DrinfeldndashSokolov reduction DSminus as usualreduction up to a shift applied to O for gκ defined with respect to Adtminusρ Iminus rather than I Iewe consider the composition

gκ -modIprime1 gκ -modAdtminusρ I

primeminus1

ins[dimNminus∆]minusminusminusminusminusminusminusminusrarr gκ -modLNψ

Explicitly let us write iminus for the Lie algebra of Iminus1 Then for ν isin hlowastκ we set

Mκ(ν) = indgκAdtminusρ L

+gind

Adtminusρ L+g

iminus Cwν

Accordingly if we write Λ isin hlowastκ for a lift of λ then by Theorem 311 and affine Skyrabin Mκ(λ) isthe image of the Verma module Mκ(Λ) of gκ under the composition

gκ -modIminus1

AvI1ψminusminusminusminusrarr gκ -modI1ψins[dimNminus∆]minusminusminusminusminusminusminusminusrarr gκ -modLNψ (96)

For any category C with a strong action of Dκ(LG) the averaging functors between the Iminus1 and(I1 ψ) invariants are adjoint up to a shift by the real dimension of N

AvI1ψlowast CIminus1 CI1ψ AvIminus1 lowast

[2 dimN ] (97)

cf the discussion in [53 sect7] In particular AvI1ψ CIminus1 rarr CI1ψ preserves compactness as it admits

a continuous right adjoint As insertion from any step of the adolescent Whittaker constructionpreserves compactness we deduce that (96) preserves compactness Finally since we are workingwith the renormalized derived category of KacndashMoody representations Mκ(Λ) is a compact object

of gκ -mod Since Iminus1 is prounipotent it is also a compact object of gκ -modIminus1 cf Proposition 421

Having shown that Mκ(λ) is compact we will calculate its dual by using (96) and Proposition422 Accordingly we need to calculate the dual of Mκ(Λ)

Lemma 98 Under the perfect pairing

Cinfin2

+lowast(g2κc L+gminusotimesminus) gκ -modotimesgminusκ+2κc -modrarr Vect

the dual of Mκ(Λ) is up to isomorphism Mminusκ+2κc(minusΛminus 2ρ)[dimN ]

Note the existing statements of the lemma in the literature written for the usual category Oie for I appear with an incorrect ρ shift [4][37][46]

Proof For a varying object V of gminusκ+2κc -mod we need to corepresent

Cinfin2

+lowast(g2κc L+gMκ(Λ)otimes V )

Cinfin2

+lowast(g2κc L+g ind

g2κc

Adtminusρ L+g

indAdtminusρ L

+g

iminus (CwΛ otimes V )) (99)

Cinfin2

+lowast(g2κc Adtminusρ L+g ind

g2κc

Adtminusρ L+g

indAdtminusρ L

+g

iminus (CwΛ otimes V ))otimes rel det(L+gAdtminusρ L+g)

The appearing relative determinant lies in cohomological degree zero as follows from consideringcancelling pairs of finite roots plusmnα and we will trivialize it

sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Since Adtminusρ L+g has no nontrivial one dimensional representations its splitting into g2κc is unique

and therefore the canonical such9 In particular we may apply Proposition 51 to rewrite Equation

9This is not true for the Lie algebra of the Iwahori i = Lie(I) and indeed the canonical section of i into g2κc isnot given by the sequence irarr L+grarr g2κc but instead differs by a factor of 2ρ on the torus This is why we brokethe induction into two steps in (99) and likely where the ρrsquos disappeared in [4] [37]


(910) as

Clowast(Adtminusρ L+g ind

Adtminusρ L+g

iminus CwΛ otimes V )

By the usual Shapiro lemma for Lie algebra cohomology we may rewrite this as

Clowast(iminusCwΛ otimes V otimes det(Adtminusρ L+giminus[1])or) Homiminus(Cminusw0Λ otimes det(Adtminusρ L

+giminus[1]) V )

Homiminus(CminuswΛ+2ρ[dimN ] V ) Homgminusκ+2κc(ind

gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g

iminus CminuswΛ+2ρ[dimN ] V )

as desired

Having calculated the dual of a Verma module for gκ we may now apply Proposition 422 andobtain a commutative diagram

gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D

gminusκ+2κc -modIminus1 c

((AvIψlowast)R)or gminusκ+2κc -modI1ψc

(911)

Recall from Equation (97) that the right adjoint (AvIψlowast)R is AvIminus1 lowast

[2 dimN ] Recalling that

AvIminus1 lowastis the composition

CI1ψlowastOblvminusminusminusrarr C

AvIminus1 lowast

minusminusminusminusrarr CIminus1

it follows from Propositions 412 415 that (AvIminus1 lowast[2 dimN ])or = AvI1ψlowast[2 dimN ] Having un-

wound the bottom horizontal arrow of Equation (911) we insert Mκ(Λ) in the top left and obtainvia Lemma 98 that

DAvI1ψlowastMκ(Λ) AvI1ψlowastMminusκ+2κc(minusΛminus 2ρ)[3 dimN ]

An application of Theorem 433 handles the subsequent insertion ins CI1ψ rarr CLNψ to give that

D ins AvI1ψlowastMκ(Λ) ins AvI1ψlowastMminusκ+2κc(minusΛminus 2ρ)[3 dimN minus 2∆] (912)

Since at any level κ and weight ν prime isin hlowast with image ν isinWfhlowast we have from Equation (96) that

ins AvI1ψlowastMκ(νprime) Mκ(ν)[minusdimN + ∆] (913)

the result follows by combining Equations (912) (913)

Example 914 Let g = sl2 and let us write k isin C for the level kκb In this situation Wk is theVirasoro vacuum algebra V irc(k) where

c(k) = 1minus 6(k + 1)2

k + 2 (915)

Then in our conventions the Drinfeld-Sokolov minus reduction of Mκ(ν) ν = vρ v isin C is theVerma module M(c(k)∆(k v)) where the conformal dimension ∆ is given by

∆(k v) =(k minus v)(k + 2minus v)

4(k + 2)minus k minus v

2 (916)

Theorem 95 applied to this case gives

DM(c(k)∆(k v)) M(c(minusk minus 4)∆(minusk minus 4minusv minus 2))[1]

Substituting in Equations (915) (916) yields the familiar form of FeiginndashFuchs duality

DM(c∆) M(26minus c 1minus∆)[1] c∆ isin C

38 GURBIR DHILLON

Remark 917 In the setting of Example 914 note that for sl2 one has Adtminusρ Iminus = I In particularone could equally well use the + reduction which sends Mκ(ν) to M(c(k)∆prime(k v)) where

∆prime(k v) =v(v + 2)

4(k + 2)minus v

2

As is visible the two differ by the Dynkin diagram automorphism of sl2

Theorem 95 reduces the classification of Verma embeddings at positive level to the analogousclassification at negative level which was done in the previous subsection We now unwind theanswer the at positive level The following propositions summarize the relevant combinatorics

Proposition 918 Let κ be negative and consider the pair of weights λ isin hlowastκ and its lsquofliprsquo minusλminus2ρ isinhlowastminusκ+2κc Then

(1) λ is antidominant if and only if minusλminus 2ρ is dominant(2) Their integral Weyl groups are the same subgroup of W ie Wλ = Wminusλminus2ρ(3) Their stabilizers in the affine Weyl group coincide ie W λ = W minusλminus2ρ

(4) Their finite integral Weyl groups are the same subgroup of Wf ie Wfλ = Wfminusλminus2ρ

Proof We refer the reader to Subsection 34 for any unfamiliar notation If we pick a lift of λ isin hlowastκto Λ isin hlowastoplusCclowastoplusCDlowast then a lift of minusλminus 2ρ is given by minusΛminus 2ρ For any coroot α isin Φor we thenhave

〈Λ + ρ α〉 = minus〈(minusΛminus 2ρ) + ρ α〉The claims of the Proposition are now straightforward as long as one remembers for (3) that theirstabilizers are generated by reflections cf the proof of Lemma 87

Corollary 919 Write hlowastκ hlowastminusκ+2κc for the map ν rarr minusν minus 2ρ Taking the orbit of λ gives anembedding WλW

λ rarr hlowastκ and similarly WminusλminusρW

minusλminus2ρ rarr hlowastminusκ+2κc Then under the identification

of the two coset spaces by Proposition 918 (2)(3) the following diagram commutes

hlowastκsim

hlowastminusκ+2κc

WλWλ

OO

Wminusλminus2ρWminusλminus2ρ

OO

Quotienting by Wf and applying Proposition 918 (4) to identify the arising double cosets thefollowing diagram commutes

Wfhlowastκsim

Wfhlowastminusκ+2κc

WfλWλWλ

OO

Wminusλminus2ρfWminusλminus2ρWminusλminus2ρ

OO(920)

We may now describe Verma embeddings at positive level κ isin κc + Qgt0κb as follows As inthe discussion at the beginning of Subsection 91 we may immediately reduce to a single blockof Category O for Wκ There is a unique highest weight in the block with a dominant lift anddwe fix such a lift λ isin hlowastκ We may accordingly parametrize the highest weights in the block byWfλWλW

λ and for a double coset w isin WfλWλW

λ we will write Mw for the corresponding

Verma moduleAs discussed in Subsection 91 WfλWλW

λ is a quotient of a Coxeter group by two parabolic

subgroups and in particular carries a Bruhat order 6 Then we have

Theorem 921 For elements v w isin WfλWλWλ Hom(MvMw) is at most one dimensional

and is nonzero if and only if v gt w


Proof This follows by combining Equation (920) Theorem 95 and Theorem 91

Finally let us note that applying the duality D within Category O for Wκ yields the classificationof homomorphisms between the co-Verma modules A(ν) ν isinWfhlowast

Corollary 922 Let κ 6= κc and fix two co-Verma modules A(micro) A(ν) Then Hom(A(micro) A(ν))is at most one dimensional and any nonzero element is a surjection

Corollary 923 At a negative level κ parametrize the co-Vermas in a block as Av v isinWfλWλWλ

where Av = DMv and Mv was defined in Subsection 91 Then Hom(Av Aw) is nonzero if andonly if v gt w

Corollary 924 At a positive level κ parametrize the co-Vermas in a block as Av v isinWfλWλWλ

where Av = DMv and Mv was defined in Subsection 92 Then Hom(Av Aw) is nonzero if andonly if v 6 w

References

[1] M Aganagic E Frenkel and A Okounkov Quantum q-Langlands correspondence Trans Moscow Math Soc791ndash83 2018

[2] Tomoyuki Arakawa Representation theory of W-algebras Inventiones mathematicae 169(2)219ndash320 2007[3] Tomoyuki Arakawa Introduction to W-algebras and their representation theory In Perspectives in Lie theory

volume 19 of Springer INdAM Ser pages 179ndash250 Springer Cham 2017[4] S Arkhipov and D Gaitsgory Localization and the long intertwining operator for representations of affine Kac-

Moody algebras Preprint httpwwwmathharvardedu~gaitsgdeGLArkhpdf 2015[5] Alexander Beilinson and Vladimir Drinfeld Chiral algebras volume 51 of American Mathematical Society Col-

loquium Publications American Mathematical Society Providence RI 2004[6] Dario Beraldo Loop group actions on categories and Whittaker invariants ProQuest LLC Ann Arbor MI 2013

Thesis (PhD)ndashUniversity of California Berkeley[7] Dario Beraldo Loop group actions on categories and Whittaker invariants Advances in Mathematics 322565ndash

636 2017[8] Dario Beraldo Wh-1 the Whittaker model httpwwwiecluniv-lorrainefr~SergeyLysenko

notes talks winter2018Wh-1(Dario)pdf 2018[9] E Bergshoeff HJ Boonstra M De Roo S Panda and A Servin On the BRST operator of W-strings Physics

Letters B 308(1-2)34ndash41 1993[10] E Bergshoeff HJ Boonstra S Panda and M De Roo A BRST analysis of W-symmetries Nuclear Physics B

411(2-3)717ndash744 1994[11] M Bershadsky W Lerche D Nemeschansky and NP Warner A BRST operator for non-critical W-strings

Physics Letters B 292(1-2)35ndash41 1992[12] Brian D Boe Daniel K Nakano and Emilie Wiesner Category O for the Virasoro algebra cohomology and

Koszulity Pacific J Math 234(1)1ndash21 2008[13] Peter Bouwknegt Jim McCarthy and Krzysztof Pilch The W3 algebra Modules semi-infinite cohomology and

BV algebras volume 42 Springer Science amp Business Media 2008[14] Peter Bouwknegt and Kareljan Schoutens W symmetry in conformal field theory Physics Reports 223(4)183ndash

276 1993[15] Jonathan Brundan Simon M Goodwin and Alexander Kleshchev Highest weight theory for finite W -algebras

Int Math Res Not IMRN (15)Art ID rnn051 53 2008[16] Justin Campbell and Gurbir Dhillon The tamely ramified Fundamental Local Equivalence at integral level

2019[17] Koos De Vos and Peter Van Driel The KazhdanndashLusztig conjecture for W algebras Journal of Mathematical

Physics 37(7)3587ndash3610 1996[18] Vinay V Deodhar Ofer Gabber and Victor Kac Structure of some categories of representations of infinite-

dimensional Lie algebras Adv in Math 45(1)92ndash116 1982[19] Boris Feigin and Edward Frenkel Semi-infinite Weil complex and the Virasoro algebra Comm Math Phys

137(3)617ndash639 1991[20] B L Feıgin Semi-infinite homology of Lie Kac-Moody and Virasoro algebras Uspekhi Mat Nauk

39(2(236))195ndash196 1984[21] B L Feıgin and D B Fuchs Verma modules over the Virasoro algebra 1060230ndash245 1984

40 GURBIR DHILLON

[22] Peter Fiebig The combinatorics of category O over symmetrizable Kac-Moody algebras Transform Groups11(1)29ndash49 2006

[23] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced Math-ematics Cambridge University Press Cambridge 2007

[24] Edward Frenkel and David Ben-Zvi Vertex algebras and algebraic curves volume 88 of Mathematical Surveysand Monographs American Mathematical Society Providence RI second edition 2004

[25] Edward Frenkel and Dennis Gaitsgory D-modules on the affine flag variety and representations of affine Kac-Moody algebras Representation Theory of the American Mathematical Society 13(22)470ndash608 2009

[26] Edward Frenkel Victor Kac and Minoru Wakimoto Characters and fusion rules for W-algebras via quantizedDrinfeld-Sokolov reduction Communications in Mathematical Physics 147(2)295ndash328 1992

[27] Igor B Frenkel and Konstantin Styrkas Modified regular representations of affine and Virasoro algebras VOAstructure and semi-infinite cohomology Advances in Mathematics 206(1)57ndash111 2006

[28] Igor B Frenkel and Anton M Zeitlin Quantum group as semi-infinite cohomology Communications in Mathe-matical Physics 297(3)687ndash732 2010

[29] Igor B Frenkel and Yongchang Zhu Vertex operator algebras associated to representations of affine and Virasoroalgebras Duke Math J 66(1)123ndash168 1992

[30] D Gaitsgory Twisted Whittaker model and factorizable sheaves Selecta Math (NS) 13(4)617ndash659 2008[31] D Gaitsgory Talk FLE-1 statement of the equivalence httpwwwiecluniv-lorrainefr~SergeyLysenko

notes talks winter2018FLE-1(Dennis)pdf 2018[32] Dennis Gaitsgory Ind-coherent sheaves arXiv preprint arXiv11054857 2011[33] Dennis Gaitsgory Notes on Geometric Langlands Generalities on DG categories httpwwwmathharvardedu

~gaitsgdeGLtextDGpdf 2012[34] Dennis Gaitsgory Kac-Moody representations (notes for day 4 talk 3) httpssitesgooglecomsite

geometriclanglands2014notes 2015[35] Dennis Gaitsgory A conjectural extension of the KazdhanndashLusztig equivalence arXiv preprint arXiv181009054

2018[36] Dennis Gaitsgory The local and global versions of the Whittaker category arXiv preprint arXiv181102468

2018[37] Dennis Gaitsgory Semi-infinite vs quantum group cohomoloy-I the negative level case http

wwwmathharvardedu~gaitsgdeGLquantum-vs-semiinfpdf 2018[38] Dennis Gaitsgory and Nick Rozenblyum A study in derived algebraic geometry Vol I Correspondences and

duality volume 221 of Mathematical Surveys and Monographs American Mathematical Society Providence RI2017

[39] Victor Ginzburg Harish-Chandra bimodules for quantized Slodowy slices Represent Theory 13236ndash271 2009[40] K Hornfeck Explicit construction of the BRST charge for W4 Physics Letters B 315(3-4)287ndash291 1993[41] CM Hull Gauging the Zamolodchikov W-algebra Physics Letters B 240(1-2)110ndash116 1990[42] V G Kac and D A Kazhdan Structure of representations with highest weight of infinite-dimensional Lie

algebras Adv in Math 34(1)97ndash108 1979[43] Victor G Kac Infinite-dimensional Lie algebras Cambridge University Press Cambridge third edition 1990[44] Masaki Kashiwara and Toshiyuki Tanisaki Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie alge-

bras III Positive rational case Asian J Math 2(4)779ndash832 1998 Mikio Sato a great Japanese mathematicianof the twentieth century

[45] Masaki Kashiwara and Toshiyuki Tanisaki Characters of irreducible modules with non-critical highest weightsover affine Lie algebras In Representations and quantizations (Shanghai 1998) pages 275ndash296 China HighEduc Press Beijing 2000

[46] Chia-Cheng Liu Semi-infinite cohomology and kazhdan-lusztig equivalence at positive level arXiv preprintarXiv180701773 2018

[47] Ivan Losev On the structure of the category O for W-algebras In Geometric methods in representation theoryII volume 24 of Semin Congr pages 353ndash370 Soc Math France Paris 2012

[48] Dragan Milicic and Wolfgang Soergel The composition series of modules induced from Whittaker modulesComment Math Helv 72(4)503ndash520 1997

[49] Robert V Moody and Arturo Pianzola Lie algebras with triangular decompositions Canadian Mathematical So-ciety Series of Monographs and Advanced Texts John Wiley amp Sons Inc New York 1995 A Wiley-IntersciencePublication

[50] CN Pope Lectures on W algebras and W gravity arXiv preprint hep-th9112076 1991[51] CN Pope LJ Romans and Kellogg S Stelle Anomaly-free W3 gravity and critical W3 strings Phys Lett B

268(IMPERIAL-TP-90-91-32)167ndash174 1991[52] Leonid Positselski Homological algebra of semimodules and semicontramodules Semi-infinite homological algebra

of associative algebraic structures volume 70 Springer Science amp Business Media 2010


[53] Sam Raskin W-algebras and Whittaker categories arXiv preprint arXiv161104937 2016[54] Sam Raskin Weak actions of Loop Groups httpwwwiecluniv-lorrainefr~SergeyLysenko

notes talks winter2018GL-5(Raskin)pdf 2018[55] Sam Raskin Homological methods in semi-infinite contexts 2019[56] Nick Rozenblyum Filtered colimits of infin-categories httpwwwmathharvardedu~gaitsgdeGL

colimitspdf 2012

[57] Vladimir Sadov On the spectra of sl(N)ksl(N)k-cosets and WN gravities Internat J Modern Phys A8(29)5115ndash5128 1993

[58] Vladimir Sadov Hamiltonian reduction of BRST complex and N=2 susy In Quantum Field Theory and StringTheory pages 305ndash333 Springer 1995

[59] K Schoutens A Sevrin and P Van Nieuwenhuizen Quantum BRST charge for quadratically nonlinear Liealgebras Communications in Mathematical Physics 124(1)87ndash103 1989

[60] C Serpe Resolution of unbounded complexes in Grothendieck categories J Pure Appl Algebra 177(1)103ndash1122003

[61] Konstantin Styrkas Quantum groups conformal field theories and duality of tensor categories Yale UniversityThesis 1998

[62] Bertrand Toen Lectures on dg-categories In Topics in algebraic and topological K-theory volume 2008 of LectureNotes in Math pages 243ndash302 Springer Berlin 2011

[63] Ben Webster Singular blocks of parabolic category O and finite W-algebras J Pure Appl Algebra 215(12)2797ndash2804 2011

[64] Emilie Wiesner The Shapovalov determinant and translation functors for the Virasoro algebra J Algebra307(2)899ndash916 2007

[65] Chuan Jie Zhu The BRST quantization of the nonlinear WB2 and W4 algebras Nuclear Phys B 418(1-2)379ndash399 1994

[66] Minxian Zhu Vertex operator algebras associated to modified regular representations of affine Lie algebrasAdvances in Mathematics 219(5)1513ndash1547 2008

[67] Minxian Zhu Regular representations of quantum groups at roots of unity International Mathematics ResearchNotices 2010(15)3039ndash3065 2010

1 Introduction


21 Categorical FeiginndashFuchs duality

22 Homomorphisms between Verma modules and the linkage principle

23 Future directions

24 Organization of the Paper

Acknowledgments

3 Preliminaries

31 The affine KacndashMoody algebra

32 The affine Weyl group and the level action

33 The affine Lie algebra at level and the level action

34 Combinatorial duality

35 The W-algebra

36 Category O

37 Duality

38 q-characters


41 Recollections on the adolescent Whittaker construction

42 Whittaker invariants of dual representations prounipotent case

43 Whittaker invariants of dual representations of loop groups


51 FeiginndashFuchs duality for Tate Lie algebras

52 FeiginndashFuchs duality for W

53 Semi-infinite cohomology for the W-algebra

54 An example chiral differential operators

6 Jordan-Houmllder content and highest weight filtrations in Category O

61 Jordan-Houmllder content

62 (Co-)Highest weight filtrations



81 The linkage principle recollections for KacndashMoody

82 The linkage principle the W-algebra

83 Some conjectures on the blocks of W

9 Verma embeddings

91 Negative level

92 Positive level

References

2 GURBIR DHILLON


Let g be a simple complex Lie algebra with triangular decomposition g = nminus oplus hoplus n1 Let G bethe simple simply connected algebraic group with Lie algebra g Fix a level κ ie an invariantbilinear form κ isin (glowast otimes glowast)G and write κc for the critical level Write gκ for the affine Lie algebraassociated to g and κ and let Wκ denote the W-algebra associated to gκ and a principal nilpotentelement of g

21 Categorical FeiginndashFuchs duality We prove a version of FeiginndashFuchs duality which ap-plies to all representations of the W-algebra Since this uses several modern notions from homo-logical algebra which may not be familiar to all readers we will approach it via several moreelementary statements A reader comfortable with the basics of stable cocomplete infin-categoriesmay wish to skip directly to Theorem 25

As mentioned in the introduction Feigin and Fuchs found a remarkable duality between theVerma modules for Virasoro at complementary central charges ie a contravariant equivalenceof categories We show this phenomenon persists for W Let us write Wκ -modhearts for its abeliancategory of representations and Vermaheartsκ for its full subcategory consisting of Verma modules Thenwe have

Theorem 21 There is a canonical equivalence of categories

Vermaheartsopκ Vermaheartsminusκ+2κc

To our knowledge Theorem 21 is new for all cases besides the Virasoro algebra A next approx-imation to our FeiginndashFuchs duality is roughly the statement that not only Homs between Vermasmatch but also Exts Let us write Wκ -modb for its bounded derived category of representations orbetter its canonical dg-enhancement If we consider Vermaκ the full pretriangulated subcategorygenerated by Vermaheartsκ we have


Vermaopκ Vermaminusκ+2κc

Another approximation of FeiginndashFuchs duality is the statement that as Verma modules areprotypical objects of tame ramification the above admits variants with wild ramification To statethis cleanly involves a slightly subtle point which we now review Let us write Wκ -modn (n fornaive) for the usual unbounded derived category of Wκ modules As objects of Wκ -modn theVerma modules are not compact However there is a slight enlargement of Wκ -modn which wedenote by Wκ -mod in which the Verma modules are now compact An analogous renormalizationwas first introduced for gκ by FrenkelndashGaitsgory and later for Wκ by Raskin [25] [53]2 Suchrenormalizations are now more broadly seen as useful facts of life in both singular finite dimensionaland smooth infinite dimensional algebraic geometry and quantizations thereof [32] [34]

Within Wκ -mod one has its subcategory of compact objects Wκ -modc We now give a heuristicsense of what these look like Recall that Zg the center of the universal enveloping algebraof g quantizes the invariant theory quotient gG Similarly the topological associative algebraassociated to W quantizes the loop space L(gG) Bounding the order of poles of an algebraic

1All results we prove have analogues for the Lie algebra of a general reductive group which may be deduced fromthe simple case Similarly one may replace our ground field C with any algebraically closed field of characteristiczero

2As a remark for experts - we in fact show all of gκ -modn embeds fully faithfully into gκ -mod cf Proposition55 Previously this was only known for the bounded below derived category and the same proof applies to W -modas well















Wfhlowast =⊔α

Pα


4 GURBIR DHILLON






















3 Preliminaries










6 GURBIR DHILLON


























κb


















ψminusrarr C


Wκ = Hinfin2



8 GURBIR DHILLON




M =oplusdisinC










DSminus Ogκ rarr O


DSminus(L(λ))





oplusdisinCM






infin2







chqM =sumdisinC

(dimMminusd)qd



chqM = chq DM









10 GURBIR DHILLON



N[NN ] prodiisinI

Uαi



ψminusrarr Ga (42)




CLNψ CLNψ (44)




















CInψ








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References















Wfhlowast =⊔α

Pα


4 GURBIR DHILLON






















3 Preliminaries










6 GURBIR DHILLON


























κb


















ψminusrarr C


Wκ = Hinfin2



8 GURBIR DHILLON




M =oplusdisinC










DSminus Ogκ rarr O


DSminus(L(λ))





oplusdisinCM






infin2







chqM =sumdisinC

(dimMminusd)qd



chqM = chq DM









10 GURBIR DHILLON



N[NN ] prodiisinI

Uαi



ψminusrarr Ga (42)




CLNψ CLNψ (44)




















CInψ








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

4 GURBIR DHILLON






















3 Preliminaries










6 GURBIR DHILLON


























κb


















ψminusrarr C


Wκ = Hinfin2



8 GURBIR DHILLON




M =oplusdisinC










DSminus Ogκ rarr O


DSminus(L(λ))





oplusdisinCM






infin2







chqM =sumdisinC

(dimMminusd)qd



chqM = chq DM









10 GURBIR DHILLON



N[NN ] prodiisinI

Uαi



ψminusrarr Ga (42)




CLNψ CLNψ (44)




















CInψ








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References






3 Preliminaries










6 GURBIR DHILLON


























κb


















ψminusrarr C


Wκ = Hinfin2



8 GURBIR DHILLON




M =oplusdisinC










DSminus Ogκ rarr O


DSminus(L(λ))





oplusdisinCM






infin2







chqM =sumdisinC

(dimMminusd)qd



chqM = chq DM









10 GURBIR DHILLON



N[NN ] prodiisinI

Uαi



ψminusrarr Ga (42)




CLNψ CLNψ (44)




















CInψ








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

6 GURBIR DHILLON


























κb


















ψminusrarr C


Wκ = Hinfin2



8 GURBIR DHILLON




M =oplusdisinC










DSminus Ogκ rarr O


DSminus(L(λ))





oplusdisinCM






infin2







chqM =sumdisinC

(dimMminusd)qd



chqM = chq DM









10 GURBIR DHILLON



N[NN ] prodiisinI

Uαi



ψminusrarr Ga (42)




CLNψ CLNψ (44)




















CInψ








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References





κb


















ψminusrarr C


Wκ = Hinfin2



8 GURBIR DHILLON




M =oplusdisinC










DSminus Ogκ rarr O


DSminus(L(λ))





oplusdisinCM






infin2







chqM =sumdisinC

(dimMminusd)qd



chqM = chq DM









10 GURBIR DHILLON



N[NN ] prodiisinI

Uαi



ψminusrarr Ga (42)




CLNψ CLNψ (44)




















CInψ








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

8 GURBIR DHILLON




M =oplusdisinC










DSminus Ogκ rarr O


DSminus(L(λ))





oplusdisinCM






infin2







chqM =sumdisinC

(dimMminusd)qd



chqM = chq DM









10 GURBIR DHILLON



N[NN ] prodiisinI

Uαi



ψminusrarr Ga (42)




CLNψ CLNψ (44)




















CInψ








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References



oplusdisinCM






infin2







chqM =sumdisinC

(dimMminusd)qd



chqM = chq DM









10 GURBIR DHILLON



N[NN ] prodiisinI

Uαi



ψminusrarr Ga (42)




CLNψ CLNψ (44)




















CInψ








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

10 GURBIR DHILLON



N[NN ] prodiisinI

Uαi



ψminusrarr Ga (42)




CLNψ CLNψ (44)




















CInψ








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References








insL CU C ins (47)





CU

Oblv

ins Oblv CU

insL

CUins Oblv CU

C C

Av

__

ins








12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

12 GURBIR DHILLON





insotimes id

Cotimes Cor Vect

CU otimes (Cor)Uδ

99






Oblv CU C Av













Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References



Vecta















Proof We compute





Then we have




D Dcop Dorc (417)

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

14 GURBIR DHILLON


















CcopF

D

Dcop

D

CorcGor Dorc






CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References



CUχcop

D

Oblv Ccop

D













Dκ(LG) -modRes

Res

Dκ(U) -mod(44)

D(U) -mod

Res


D(U prime) -mod






insotimes ins

Cotimes Cor


(426)


16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

16 GURBIR DHILLON




CInψ






( limminusrarrFnm


Corn


Cn otimes Corn

ampamp

insnotimes insn


(429)



















CorInminusψ





limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References




limminusrarriisinI


iisinICi)

c (431)



limminusrarrFα

Ci limlarrminusGα

Ci










CInψcop CcopLNψC


inmlowast

CorInminusψc




CorInminusψc

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

18 GURBIR DHILLON















Cinfin2





Cinfin2







Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References




Cinfin2












L0M




Spj otimes F k indM















20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

20 GURBIR DHILLON







Cinfin2




Cinfin2











Cinfin2




















Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References







Hom(γLc d)sim

(58)


(59)



〈c d〉 Cinfin2







Cinfin2







Cinfin2



write Wψκ = H


Cinfin2


22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

22 GURBIR DHILLON





minusψminusκ+2κc

-mod (514)













CUj ψ


CUj ψOblv

δhminus

CUiψ

δhminus

CUiψAvψlowast

δhminus

CUj ψ

δhminus


Avhψlowast CUj hψ

(519)




ψprime

κprime -mod






CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References



CLNψ(44)

(516)

CLNψ

(516)

CLNhψ(44) CLNhψ










CInhψ




CI+n ψ

δhminus

(1)


CInψ

(2)δhminus

Oblvoo[minus]

limminusrarrinm

CInψ

(3)δhminus

simlimlarrminusinm

CInψ

(4)δhminus

limlarrminusOblv

CI+n ψ

δhminus

Avψlowastoo



inmlowast

CInhψOblvoo


CInhψsim

limlarrminusinm


CI+n hψ

Avhψlowastoo



CIj ψ

δhminus

ins limminusrarrinm

CInψ

δhminus

simlimlarrminusinm

CInψ

δhminus

ev CIkψ

δhminus


inm

CInhψsim

limlarrminusinm

CInhψev CIkhψ



24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

24 GURBIR DHILLON




Wψκ -mod+ Whψ

κ -mod+ (521)


Wψκ -mod Whψ

κ -mod (522)



gκ -modLNψ(513)

(516)

Wψκ -mod

(522)

gκ -modLNhψ(513)

Whψκ -mod




Cinfin2







c(κ) = dim gk






k + hor









Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References




Cinfin2







CDO oplusλisinP+



Cinfin2


oplusλisinP+






Cinfin2











26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

26 GURBIR DHILLON






Mγ =oplusdisinγ+Z

Md


γisinCZMγ























Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References











Mi = M






Ai = M






28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

28 GURBIR DHILLON


Wfhlowast =⊔α

Pα


























We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References


We deduce





Mi rarr 0














oplusαMiα By


M limminusrarri

Mi limminusrarri

oplusα

Miα oplusα

limminusrarri

Miα






30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

30 GURBIR DHILLON














































32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

























32 GURBIR DHILLON














as desired










λ






Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References




Omonλsimι

DSminus

Omonλ

T

OWκλsim

Omonminusρ





9 Verma embeddings




λ For a



34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

34 GURBIR DHILLON











λ













P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References



P = C[W i(n)] 1 6 i 6 rk g n gt 0




















36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

36 GURBIR DHILLON



primeminus1



primeminus1




+gind

Adtminusρ L+g

iminus Cwν


gκ -modIminus1




[2 dimN ] (97)






Cinfin2





Cinfin2


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g


Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g



sim= Cinfin2


g2κc

Adtminusρ L+g

indAdtminusρ L

+g






(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References


(910) as


Adtminusρ L+g




+giminus[1]) V )


gminusκ+2κc

Adtminusρ L+g

indAdtminusρ L

+g


as desired


gκ -modIminus1 cop

AvI1ψlowast

D

gκ -modI1ψcop

D



(911)





AvIminus1 lowast











c(k) = 1minus 6(k + 1)2

k + 2 (915)




2 (916)





38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

38 GURBIR DHILLON



4(k + 2)minus v

2












hlowastκsim

hlowastminusκ+2κc

WλWλ

OO


OO


Wfhlowastκsim


WfλWλWλ

OO


OO(920)

















References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References









References





















40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

40 GURBIR DHILLON































colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References




colimitspdf 2012












1 Introduction






Acknowledgments

3 Preliminaries





35 The W-algebra

36 Category O

37 Duality

38 q-characters


















9 Verma embeddings

91 Negative level

92 Positive level

References

SEMI-INFINITE COHOMOLOGY AND THE LINKAGE PRINCIPLE …gsd/semiinflinkage.pdf · SEMI-INFINITE...

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