ASYMPTOTICS OF GEOMETRIC WHITTAKER...

ASYMPTOTICS OF GEOMETRIC WHITTAKER COEFFICIENTS

D. ARINKIN AND D. GAITSGORY

Introduction

Date: March 16, 2015.

1

2 D. ARINKIN AND D. GAITSGORY

Part I: The Key Calculation

1. Statement of the problem

1.1. The geometric objects.

1.1.1. Recall the stack BunB defined as in [BG1]. Recall that BunB admits a decompositioninto locally closed substacks, parameterized by elements λ ∈ Λpos,

BunB =⋃

λ∈Λpos

Bun=λ

B .

For each λ ∈ Λpos we let

λ : Bun=λ

B → BunB

denote the corresponding locally closed embedding.

We have a canonical identification

Bun=λ

B ' Xλ × BunB .

We let Bun≤λB denote the open substack of BunB equal to⋃

0≤µ≤λ

Bun=µ

B .

1.1.2. We let

p : BunB → BunG and q : BunB → BunT

denote the canonical projections, and by

p : BunB → BunG and q : BunB → BunT

their restrictions to BunB ⊂ BunB , respectively.

Note that we have a commutative diagram

Xλ × BunB∼−−−−→ Bun

=λ

B

id×qy yqλ

Xλ × BunT −−−−→ BunT ,

where the bottom horizontal arrow is the map

D,PT 7→ PT (−D),

where D ∈ Xλ is a Λpos-colored divisor.

ASYMPTOTICS OF GEOMETRIC WHITTAKER COEFFICIENTS 3

1.1.3. Let PT be a T -bundle. We let BunN,PT denote the fiber product

BunB ×BunT

pt,

where the map BunB → BunT is q and pt→ BunT is given by PT .

The group T acts by automorphisms of PT ; by functoriality, we obtain a T -action onBunN,PT . The quotient stack BunN,PT /T identifies with

BunB ×BunT

pt /T ;

in particular, the resulting map

BunN,PT /T → BunB

is a closed embedding.

We let r denote the tautological projection BunN,PT → BunN,PT /T .

The stack BunN,PT inherits a decomposition into locally closed substacks with

Bun=λ

N,PT ' Xλ ×

BunTBunB ,

where the map Xλ → BunT is

D 7→ PT (D).

By a slight abuse of notation, we shall use the same symbol λ to denote also the maps

Bun=λ

N,PT → BunN,PT and Bun=λ

N,PT /T → BunN,PT /T,

respectively.

Similarly, by a slight abuse of notation we will denote by id×q also the maps

Bun=λ

N,PT ' Xλ ×

BunTBunB

id×q−→ Xλ ×BunT

BunT = Xλ

and

(1.1) Bun=λ

N,PT /T ' (Xλ ×BunT

BunB)/Tid×q−→ (Xλ ×

BunTBunT )/T = Xλ × pt /T,

respectively.

1.2. The basic character sheaf.

1.2.1. In what follows we fix one and for all a line bundle ω12

X equipped with an isomorphsim

(ω12

X)⊗2 ' ωX .

We let ρ(ωX) denote the T -bundle on X induced from the line bundle ω12

X by means of thecocharacter 2ρ : Gm → T .

Consider the stack

BunN,ρ(ωX) .

As in [FGV], there exists a canonical map

ev : BunN,ρ(ωX) → A1.


1.2.2. The map ev is constructed as follows. Consider the map

B → B/[N,N ]ZG ' Πi

(Gm nGa),

where i runs through the set of vertices of the Dynkin diagram of G. From here we obtain amap of stacks

BunN,ρ(ωX) → Πi

(BunGmnGa ×

BunGm

pt

),

where pt→ BunGm is given by the line bundle ωX .

Now, the stack BunGmnGa ×BunGm

pt classfies short exact sequences

0→ ωX → E→ OX → 0,

and hence is equipped with a canonocal map to A1.

Finally, we let ev be the composition

(1.2) BunN,ρ(ωX) → Πi

(BunGmnGa ×

BunGm

pt

)→ Π

iA1 sum−→ A1.

1.2.3. Setψ := ev∗(A-Sch),

where A-Sch ∈ D-mod(A1) is the algebraic-Schreier sheaf (i.e., the exponential D-module,shifted cohomologically by [−1]).

Setψ := (0)!(ψ) ∈ D-mod(BunN,ρ(ωX)).

Remark 1.2.4. As was observed in [FGV], the extension ψ ψ is clean, i.e., the canonical map

(1.3) (0)!(ψ)→ (0)∗(ψ)

is an isomorphism.

1.2.5. Consider the objectr!(ψ) ∈ D-mod(BunN,ρ(ωX)/T ).

For each λ ∈ Λpos consider

(λ)! r!(ψ) ∈ D-mod(Bun=λ

N,ρ(ωX)/T ).

Consider now the object

(1.4) Mλ := (id×q)∗ (λ)! r!(ψ) ∈ D-mod(Xλ × pt /T ),

where id×q : Bun=λ

N,ρ(ωX)/T → Xλ × pt /T is as in (1.1).

1.2.6. The goal of Part I of this paper is concerned with the object Mλ of (1.4). On the onehand, in Theorem 1.3.6, we will calculate Mλ more or less explicitly.

On the other hand, and which is more important, in Theorem 1.4.6, we will relate Mλ to thecalculation of a certain Constant Term functor.

The proofs of Theorems 1.3.6 and 1.4.6 will share a common core, given by Theorem 2.2.3.The latter theorem (like most of the assertions of this kind) is a corollary of Braden’s theoremabout hyperbolic restrictions; the deduction will be the subject of Part II of the paper.

In Part III of the paper we will explain how Theorem 1.4.6 fits into the geometric Langlandsprogram.


1.2.7. Example. Let us describe explicitly the object M0, i.e., Mλ for λ = 0. This is easy to do“by hand”. Namely, we claim that M0 ∈ D-mod(pt /T ) identifies with Hc(T ), where the latteris the direct image with compact supports of k ∈ Vect = D-mod(pt) under the map

triv : pt→ pt /T.

Indeed, note that in the composition (1.2), the first map is a smooth unipoten gerbe (see[DrGa1] for what this means), so the functors of *-pullback and *-pushforward define mutuallyinverse equivalences of categories.

Hence, M0 identifies with the direct image under Ar/T → pt /T of the direct image withcompact supports under Ar → Ar/T of

A-Schr ∈ D-mod(Ar),

where r denotes the semi-simple rank of G.

Hence, M0 identifies with the direct image under Ar/T → pt /T of the direct image withcompact supports under Ar → Ar/T of

A-Schr ∈ D-mod(Ar).

By the contraction principle (see [DrGa2]), the operation of direct image under Ar/T → pt /Tcan be replaced by that of *-pullback under pt /T → Ar/T .

The stated answer for M0 follows now by base change from the Cartesian square

pt −−−−→ Ar

triv

y ypt /T −−−−→ Ar/T.

1.3. The sheaves Ωλ.

1.3.1. In this subsection we recall the objects

Ωλ ∈ D-mod(Xλ)♥, λ ∈ Λpos,

introduced in [BG2].

The object Ωλ is characterised by the requirement that its !-fiber at

D = Σiλi · xi ∈ Xλ, xi 6= xj ,

identifies with ⊗i

(C·(n))−λi ,

where C·(n) is the cohomological Chevalley complex of the Lie algebra n, and superscript −λimeans the (−λi)-graded component with respect to the adjoint action of T . The sheaf structureon Ωλ is given by the structure on C·(n) of commutative DG algebra.


1.3.2. Examples. For G = SL2 and λ = n ∈ Z≥0 (so that Xλ is the n-th symmetric power

X(n)), the sheaf Ωλ is the (clean) extension of the sign local system onX(n).

The same happens for any G for λ = n · αi, where αi is a simple coroot.

Consider now G = SL3 with its two simple roots α1 and α2. For λ = α1 + α2 we haveXλ = X ×X, and we have

Ωα1+α2 = j∗(kX kX)[2],

where j : X ×X −∆(X) → X ×X.

1.3.3. Although this will not be necessary for the sequel, let us add a few more pieces ofinformation regarding the sheaves Ωλ.

First, the assigment

λ 7→ Ωλ

has a factorization property:

(1.5) Ωλ1+λ2 |(Xλ1×Xλ2 )disj' Ωλ1 Ωλ2 |(Xλ1×Xλ2 )disj

,

where

(Xλ1 ×Xλ2)disj ⊂ Xλ1 ×Xλ2

is the open subset corresponding to pairs

(D1, D2), supp(D1) ∩ supp(D2) = ∅.

Note that the addition map

Xλ1 ×Xλ2 → Xλ1+λ2

is etale when retsricted to (Xλ1 ×Xλ2)disj, so in the left-hand side of (1.5) we can take eitherthe ! or the *-retsriction.

1.3.4. The isomorphism (1.5) allows to describe the sheaves Ωλ inductively. Indeed, suppose we

know Ωλ for λ′ < λ. Then (1.5) implies that we know the restriction of Ωλ to Xλ−∆(X)j→ Xλ.

According to [BG2, Proposition 3.2], we have:

Ωλ =

kX [1] if λ is a simple coroot,

j!∗(Ωλ|Xλ−∆(X)) if ∃w ∈W, λ = ρ− w(ρ) with `(w) = 2,

j!∗(Ωλ|Xλ−∆(X)) ' H0(j∗(Ω

λ|Xλ−∆(X))), if ∃w ∈W, λ = ρ− w(ρ) with `(w) ≥ 3,

H0(j∗(Ωλ|Xλ−∆(X))) ' j∗(Ωλ|Xλ−∆(X)) if 6 ∃w ∈W, λ = ρ− w(ρ),

where H0 referes to taking cohomology with respect to the D-module (i.e., perverse) t-structure.

1.3.5. The first main result of Part I of this paper reads:

Theorem 1.3.6. There exists a canonical isomorphism in D-mod(Xλ × pt /T )

Mλ ' Ωλ[−2|λ|]Hc(T ),

where |λ| is the length of λ.

1.4. Relation to the Constant Term functor(s).


1.4.1. Recall that the map

BunN,ρ(ωX)/T → BunB

is a closed embedding.

By a slight abuse of notation we shall identify objects of D-mod(BunN,ρ(ωX)/T ) with the

corresponding objects of D-mod(BunB).


Bun=λ

N,ρ(ωX)/T −−−−→ Bun=λ

Bid×q−−−−→ Xλ × BunT

∼y ypXλ×id

(Xλ ×BunT

BunB)/T −−−−→ Xλ × pt /TAJ−−−−→ BunT ,

where AJ is a version of the Abel-Jacobi map,

D 7→ ρ(ωX)(D),

and where pXλ : Xλ → pt.

1.4.2. Let W ∈ D-mod(BunG) denote the direct image of ψ ∈ D-mod(BunN,ρ(ωX)) with compactsupports under the forgetful map

BunN,ρ(ωX) → BunG .

Tautologically,

W ' p! r!(ψ).

Sometimes, the object W ∈ D-mod(BunG) goes under the same the first Whittaker coefficient.

1.4.3. Recall the Constant Term functor

CT∗ := q∗ p! : D-mod(BunG)→ D-mod(BunT ),

see [DrGa4].

Recall the object Mλ ∈ D-mod(Xλ × pt /T ), see Sect. 1.4. We claim that there exists acanonically defined map

(1.6) AJ!(Mλ)→ CT∗(W).

Indeed, let us write

AJ!(Mλ) = (pXλ × id)! (id×q)∗ (λ)! r!(ψ) and W = p! r!(ψ),

and we claim that there is a natural transformation

(1.7) (pXλ × id)! (id×q)∗ (λ)! → CT∗ p!.


1.4.4. Namely, consider the unit of the adjunction

Id→ p! p!,

and apply to both sides the functor

(pXλ × id)! (id×q)∗ (λ)! : D-mod(BunB)→ D-mod(BunT ).

We obtain a map

(1.8) (pXλ × id)! (id×q)∗ (λ)! → (pXλ × id)! (id×q)∗ (λ)! p! p! '

' (pXλ × id)! (id×q)∗ (p λ

)! p!.

So, it is enough to construct a natural transformation

(1.9) (pXλ × id)! (id×q)∗ (p λ

)! → CT∗ .

However,

p λ = p (pXλ × id) : Xλ × BunB → BunG,

so the expression in (1.9) is isomorphic to

(1.10) (pXλ × id)! (id×q)∗ (pXλ × id)! p!.

Further, from the Cartesian diagram

Xλ × BunBpXλ×id

−−−−−→ BunB

id×qy yq

Xλ × BunTpXλ×id

−−−−−→ BunT

we obtain a natural transformation (in fact, an isomorphism, since the horizontal maps areproper)

(pXλ × id)! (id×q)∗ → q∗ (pXλ × id)!.

Hence, the expression in (1.10) maps (in fact, isomorphically) to

(1.11) q∗ (pXλ × id)! (pXλ × id)! p!.

Finally, applying the co-unit of the adjunction (pXλ × id)! (pXλ × id)! → Id, we obtain thatthe expression in (1.11) maps to

q∗ p! = CT∗,

as desired.

1.4.5. The main result of Part I of this paper is:

Theorem 1.4.6. The map (1.6) is an isomorphism.

2. Calculation via the Zastava spaces

2.1. Zastava spaces: recollections.


2.1.1. Let ZλPT denote the Zastava space corresponding to PT ∈ BunT .

I.e., ZλPT is the open substack(BunN,PT ×

BunGBunB−,λ+deg(PT )

)gen.trans

⊂ BunN,PT ×BunG

BunB−,λ+deg(PT ),

corresponding to B- and B−-reductions that are transversal at the generic point of the curve.

In the above formula BunB−,λ+deg(PT ) is the connected component of BunB− correspondingto B-bundles, for which the induced T -bundle has degree λ+ deg(PT ).

Remark 2.1.2. A basic feature of ZλPT is that it is actually a (quasi-projective) scheme.

2.1.3. We let′p− : ZλPT → BunN,PT

denote the tautological projection.

For 0 ≤ µ ≤ λ, we let ′µ denote the locally closed embedding

Zλ,=µPT:= ZλPT ×

BunN,PT

Bun=µ

N,PT → ZλPT .

In particular, for µ = 0 we letZλPT ⊂ Z

λPT

denote the corresponding open subscheme.

2.1.4. According to [BFGM], there exists a canonical projection

πλ : ZλPT → Xλ

that makes the diagram

ZλPT −−−−→ BunB−,λ+deg(PT )

πλ

y yXλ −−−−→ BunT

commute, where the bottom horizontal arrow is the map D 7→ PT (D).

It is known that the map πλ induces an isomorphism between Zλ,=λPTand Xλ. Hence, the

map ′λ provides a section of the projection πλ. We shall also use the notation

sλ := ′λ.

The composed map

Xλ sλ' Zλ,=λPT

′p−−→ Bun=λ

N,PT = Xλ ×BunT

BunB

equals the map ιλ.

2.2. An adaptation of Braden’s theorem.


2.2.1. For any PT consider the diagram

(2.1)

Xλ × pt /Tsλ−−−−→ ZλPT /T

πλ−−−−→ Xλ × pt /T

ιλ

y y′p−(Xλ ×

BunTBunB)/T

λ−−−−→ BunN,PT /T

id×qy

Xλ × pt /T,

in which the inner square is Cartesian and the composite maps are both equal to idXλ×pt /T .

By base change, we have an isomorphism of functors

(2.2) (id×q)∗ (λ)! (′p−)∗ (πλ)! ' IdD-mod(Xλ×pt /T ),

from which we obtain a natural transformation of functors

(2.3) (id×q)∗ (λ)! → (πλ)! (′p−)∗, D-mod(BunN,PT /T )→ D-mod(Xλ × pt /T ).

2.2.2. We will apply Braden’s theorem (see [DrGa3]) to prove:

Theorem 2.2.3. The natural transformation (2.3) is an isomorphism.

Theorem 2.2.3 will be proved in Sect. 5.1.

2.3. Proof of Theorem 1.3.6. In this subsection we will show how Theorem 2.2.3 impliesTheorem 1.3.6.

2.3.1. Take PT = ρ(ωX), and consider the corresponding Zastava space Zλρ(ωX). Recall the map

′p− : Zλρ(ωX) → BunN,ρ(ωX).

The key ingredient that connects Mλ with Ωλ is provided by the followiing theorem (see[Ras]):

Theorem 2.3.2. There exists a canonical isomorphism

πλ! (′p−)∗(ψ) ' Ωλ[−2|λ|].

2.3.3. Thus, in order to prove Theorem 1.3.6, we need to construct a canonical isomorphism

(2.4) (id×q)∗ (λ)! r!(ψ) ' πλ! (′p−)∗(ψ)Hc(T )

of objects of D-mod(Xλ × pt /T ).

Let us apply the natural isomorphism (2.3) to

r!(ψ) ∈ D-mod(BunN,ρ(ωX)/T ).

We obtain that the left-hand side identifies with

(πλ)! (′p−)∗ r!(ψ).


Applying base change along the lower square in the diagram

Xλ id× triv−−−−−→ Xλ × pt /T

πλ

x xπλZλPT

′r−−−−→ ZλPT /T

′p−y y′p−

BunN,ρ(ωX)r−−−−→ BunN,ρ(ωX)/T,

we rewrite

(πλ)! (′p−)∗ r!(ψ) ' (πλ)! (′r)! (′p−)∗(ψ) ' (id× triv)! (πλ)! (′p−)∗(ψ) '

' (πλ)! (′p−)∗(ψ)Hc(T ),

as required.

2.4. A reduction step towards the proof of Theorem 2.2.3. As was mentioned earlier,Theorem 2.2.3 will be deduced from Braden’s theorem. We would like to apply Braden’stheorem to the algebraic stack BunN,PT . The problem is that we cannot quite do this, becauseBraden’s theorem is only known for algebraic spaces, and not general algebraic stacks.

In this subsection we will show that the assertion of Theorem 2.2.3 can be formally deduced

from the case when the pair (PT , λ) is such that Bun≤λN,PT is an algebraic space, in which case

Braden’s theorem can be applied.

That said, we should say that, given Theorem 5.1.3 established in Part II of the paper, wecan reprove an analog of Braden’s theorem specifically for the stack BunN,PT , using the methodof [DrGa3]. I.e., the reduction to the case of algebraic spaces is not really necessary.

2.4.1. Let us be given a diagram of algebraic stacks

(2.5)

Y0 ι−−−−−→ Y−q−−−−−→ Y0

ι+

y yp−

Y+ p+

−−−−→ Y

q+

yY0,

whereq+ ι+ = idY− = q− ι−.

and where the mapY0 → Y+ ×

YY−

is an open embedding.

Then as in (2.2), we obtain a natural transformation

q+∗ (p+)! (p−)∗ (q−)! → IdD-mod(Y0),

from which we obtain a natural transformation

(2.6) (q+)∗ (p+)! → (q−)! (p−)∗.


2.4.2. We wish to know when for a given

F ∈ D-mod(Y)

the resulting map (q+)∗ (p+)!(F)→ (q+)∗ (p+)!(F) is an isomorphism.

Suppose that we are given a similar diagram

(2.7)

Y0 ι−−−−−→ Y−q−−−−−→ Y0

ι+

y yp−

Y+ p+

−−−−→ Y

q+

yY0.

2.4.3. Suppose that we are given a map from the diagram (2.7) to the diagram (2.5), such that:

• The maps φ− : Y− → Y− and φ0 : Y0 → Y0 are isomorphisms.;

• The maps φ : Y→ Y and φ+ : Y+ → Y+ are smooth;• The diagram

Y+ p+

−−−−→ Y

φ+

y yφY+ p+

−−−−→ Yis Cartesian.

The following is an easy particular case of [DrGa4, Theorem 4.3.4]

Lemma 2.4.4. Let F ∈ D-mod(Y) be such that for F := φ∗(F) the map

(q+)∗ (p+)!(F)→ (q+)∗ (p+)!(F)

is an isomorphism. Assume also that the maps

(2.8) (q+)∗ (p+)!(F)→ (q+)∗ (ι+)∗ (ι+)∗ (p+)!(F) ' (ι+)∗ (p+)!(F)

and

(2.9) (q+)∗ (p+)!(F)→ (q+)∗ (ι+)∗ (ι+)∗ (p+)!(F) ' (ι+)∗ (p+)!(F)

are isomorphisms. Then the map

(q+)∗ (p+)!(F)→ (q+)∗ (p+)!(F)

is an isomorphism.

Proof. Diagram chase shows that we have the following commutative diagram of natural trans-formations

(φ0)∗ (q+)∗ (p+)! (φ0)∗(2.6)−−−−−−→ (φ0)∗ (q−)! (p−)∗y y(q+)∗ (φ+)∗ (p+)! (q−)! (φ−)∗ (p−)∗y y∼

(q+)∗ (p+)! φ∗ (2.6)(φ∗)−−−−−−→ (q−)! (p−)∗ φ∗.


We need to show that the top horizontal arrow in this diagram is an isomorphism whenevaluated on F. We shall do so by showing that all other maps are isomorphisms.

The assumption on the map of diagrams implies that the lower left vertical arrow and theupper right vertical arrows are isomorphisms. The bottom horizontal arrow is an isomorphismby assumption. Hence, it remains to show that the upper left vertical arrow is an isomorphism.

However, this follows from the commutative diagram

(φ0)∗ (q+)∗ (p+)! −−−−→ (φ0)∗ (ι+)∗ (p+)!y y∼(q+)∗ (φ+)∗ (p+)! −−−−→ (ι+)∗ (φ+)∗ (p+)!

and the assumption that the maps (2.8) and (2.9) are isomorphisms.

2.4.5. Let us first take the diagram (2.5) to be (2.1). To prove Theorem 2.2.3 we have to showthat the corresponding natural transformation (2.6) is an isomorphism.

For a point x ∈ X , let

BungoodxN,PT ⊂ BunN,PT

be the open substack, where we do not allow the generalized B-reduction to degenerate atx ∈ X.

Denote also

Bun≤λ,goodxN,PT and Bun

=λ,goodxN,PT

the corresponding open substacks of Bun≤λN,PT and Bun

=λ

N,PT , respectively. We have

Bun=λ,goodxN,PT ' (X − x)λ ×

BunTBunB .

It is enough to show that the natural transformation (2.6) is an isomorphism for the diagram

(2.10)

(X − x)λ × pt /T −−−−→ ZλPT /T ×Xλ

(X − x)λ −−−−→ (X − x)λy yBun

=λ,goodxN,PT /T −−−−→ Bun

≤λ,goodxN,PT /Ty

(X − x)λ

for any x. We shall do so by applying Lemma 2.4.4.

Note that the natural transformation (2.8) is an isomorphism, by the contraction principle,see [DrGa3].


2.4.6. Let µ be a dominant coweight, and set P′T := PT (−µ · x). We claim that there exists acanonically defined commutative (in fact Cartesian) diagram

Bun=λ,goodxN,P′T

−−−−→ BungoodxN,P′Ty y

Bun=λ,goodxN,PT −−−−→ Bun

goodxN,PT ,

in which the vertical arrows are smooth.

Indeed, let BunB,PT (Dx) (resp., BunB,P′T (Dx)) denote the stack of B-bundles on the formal

disc around x in X, equipped with an identification of the induced T -bundle with PT |Dx(resp.,

P′T |Dx).

We have canonically defined maps

BungoodxN,PT → BunB,PT (Dx) and Bun

goodxN,P′T

→ BunB,P′T (Dx).

We also have a smooth map

BunB,P′T (Dx)→ BunB,PT (Dx),

and a Cartesian square

BungoodxN,P′T

−−−−→ BunB,P′T (Dx)y yBun

goodxN,PT −−−−→ BunB,PT (Dx).

2.4.7. Note also that there is a canonical isomorphism

ZλPT ×Xλ

(X − x)λ ' ZλP′T ×Xλ(X − x)λ

(the Zastava space is “local” in X).

By construction, the diagram

BungoodxN,P′T

′p−←−−−− ZλP′T×Xλ

(X − x)λy yBun

goodxN,PT

′p−←−−−− ZλPT ×Xλ

(X − x)λ

commutes as well.

Hence, we obtain a map from the diagram

(2.11)

(X − x)λ × pt /T −−−−→ ZλP′T/T ×

Xλ(X − x)λ −−−−→ (X − x)λy y

Bun=λ,goodxN,P′T

/T −−−−→ Bun≤λ,goodxN,P′T

/Ty(X − x)λ

to that in (2.11).


2.4.8. Applying Lemma 2.4.4, we obtain that the assertion of Theorem 2.2.3 for a given pairPT and λ follows from the validity of Theorem 2.2.3 for P′T := PT (−µ · x) with µ ∈ Λ+, andthe same λ for all x ∈ X.

Choose µ so that deg(PT )+λ−µ is anti-dominant and regular. Note that in this case points

of Bun≤λN,P′T

admit no non-trivial automorphisms, so the open substack Bun≤λN,P′T

is an algebraicspace.

Hence, we obtain that assertion of Theorem 2.2.3 for a given λ follows from the case when

PT is such that Bun≤λN,PT is an algebraic space.

3. Proof of Theorem 1.4.6

3.1. Compatibility between diagrams.

3.1.1. Let us be given be a pair of diagrams (2.5) and (2.7) and a map between them. Thenthe construction in Sect. 1.4.4 gives rise to a natural transformation

(3.1) (φ0)! (q+)∗ (p+)! → (q+)∗ (p+)! φ!.

Let us make the following assumptions:

• The diagram

(3.2)

Y0 ι+−−−−→ Y+

φ0

y yφ+

Y0 ι+−−−−→ Y+

is Cartesian;

• The map Y− → Y×YY− is an open embedding.

In particular, we obtain a natural transformation

(3.3) (φ−)! (p−)∗ → (p−)∗ φ!.

Furthermore, diagram chase shows that the following diagram of natural transformations iscommutative:

(3.4)

(φ0)! (q+)∗ (p+)! (3.1)−−−−→ (q+)∗ (p+)! φ!

(φ0)!(2.6)

y(φ0)! (q−)! (p−)∗

y(2.6)(φ!)

∼y

(q−)! (φ−)! (p−)∗(q−)!(3.3)−−−−−−→ (q−)! (p−)∗ φ!.


3.1.2. We apply the commutative diagram (3.4) in the following situation. We take (2.7) to bethe diagram

Xλ × pt /Tsλ−−−−→ ZλPT /T

πλ−−−−→ Xλ × pt /T

ιλ

y y′p−(Xλ ×

BunTBunB)/T

λ−−−−→ BunN,PT /T

id×qy

Xλ × pt /T,

of (2.1).

We take the diagram (2.5) to be the diagram

BunT −−−−→ BunB−q−−−−−→ BunTy yp−

BunBp−−−−→ BunG

q

yBunT .

We evaluate the functors in (3.4) on the object

r!(ψ) ∈ D-mod(BunB).

The statement of Theorem 1.4.6 says that the top horizontal arrow in the resulting diagram(3.4) is an isomorphism. We shall do so by showing that all other arrows are isomorphisms.

3.1.3. The fact that the upper left vertical arrow is an isomorphism is the content of Theo-rem 2.2.3. The fact that the right vertical arrow is an isomorphism follows from the fact thatthe natural transformation

q∗ p! → (q−)! (p−)∗

is an isomorphism, see [DrGa4].

3.1.4. Thus, it remains to show that the natural transformation

(q−)! (φ−)! (p−)∗ → (q−)! (φ−)! (p−)∗,

induced by (3.3), yields an isomorphism when evaluated on the object

r!(ψ) ∈ D-mod(BunN,PT /T ).

Concretely, we need to show the following. Consider the open embedding(BunN,ρ(ωX) ×

BunGBunB−

)gen.trans.j→(

BunN,ρ(ωX) ×BunG

BunB−

).

We need to show that the map

j! j∗ (′p−)∗(ψ)→ (′p−)∗(ψ)

induces an isomorphism after taking the direct image with compact supports along the projec-tion

BunN,ρ(ωX) ×BunG

BunB− → BunB−q−−→ BunT .


3.1.5. The stack BunN,ρ(ωX) ×BunG

BunB− admits a decomposition into locally closed pieces in-

dexed by the Weyl group:(BunN,ρ(ωX) ×

BunGBunB−

)=⋃w

(BunN,ρ(ωX) ×

BunGBunB−

)w,

where the open piece

(BunN,ρ(ωX) ×

BunGBunB−

)gen.trans.

corresponds to w = 1.

We will show that for each w 6= 1, the direct image with compact supports under the map

(3.5) Zw :=

(BunN,ρ(ωX) ×

BunGBunB−

)w→ BunB−

q−−→ BunT

of the *-restriction of ψ, vanishes.

3.2. Calculation for w 6= 1.

3.2.1. The stack Zw maps to

XΛpos

:=⊔

µ∈Λpos

Xµ,

see [BG2, Sect. 10.9]. Let us denote this map by πw.

The map (3.5) is the composition of the map πw, followed by the Abel-Jacobi map

XΛpos

→ BunT , D 7→ ρ(ωX)(w(D)).

We will show that the object

(πw)!(ψ|Zw) ∈ D-mod(XΛpos

)

vanishes.

For that it suffices to show that (πw)!(ψ|Zw) vanishes, when restricted to (X−x)Λpos

for anyx ∈ X.

3.2.2. Let Bw denote the subgroup B ∩ Adw(B−). It is equipped with a canonical projectionto T . Let Nw denote the kernel of this projection, i.e., Nw = N ∩Adw(B−).

For a test-scheme S and a map z : S → Zw let D denote the resulting map S → XΛpos

. Thepoint z corresponds to a G-bundle PG on S ×X, equipped with a reduction α to B (for whichthe induced T -bundle is ω(ρ)), and a reduction β to B−, which are in position w at the genericpoint of X.

By the construction of the map πw (see [BG2, Sect. 10.9]), the data of (PG, α, β), whenrestricted to the open subset

U := S ×X −GraphD,

is induced from a uniquely defined Bw-bundle PBw , for which the induced T -bundle is identifiedwith ρ(ωX).

Let us denoteZw,goodx := Zw ×

XΛpos(X − x)Λpos

,

and let us choose a trivialization of the T -bundle ρ(ωX) over the formal disc Dx around x inX.

Thus, we obtain a map of stacks

Zw,goodx → BunNw(Dx) ' pt /Nw(Dx),


where Dx denotes the formal disc around x, and where Nw(Dx) denotes the group-schemeclassifying maps Dx → Nw.

3.2.3. Consider the fiber product

Zw,levelx := Zw,goodx ×pt /Nw(Dx)

pt .

This is the moduli stack of (PG, α, β, ε), where (PG, α, β) are as above, and ε is the datumof trivialization of PNw over Dx. One can show that Zw,levelx is in fact a scheme (of infinitetype).

The stack (scheme) Zw,levelx is acted on by the group ind-scheme Nw(D×x ), where Nw(D×x )classifies maps from the formal punctured disc D×x to Nw. This action preserves the map

Zw,levelx → (X − x)Λpos

,

For any group-subscheme N′ ⊂ N := Nw(D×x ), we can find a group-subscheme N′′ ⊂ N0 :=Nw(Dx) of finite codimension, which is normal in N′. Thus, we obtain an action of the (finite-dimensional) unipotent group N′/N′′ on the stack (of finite type)

Zw,levelx/N′′.

This action preserves the projection

(3.6) Zw,levelx/N′′ → Zw,levelx/N0 = Zw,levelx → (X − x)Λpos

.

3.2.4. Consider the canonical projection

N → Gra.

Consider the map

N(D×x )→ Gra(D×x )res−→ Gra

sum−→ Ga,

where the map

res : Gra(D×x )→ Grais defined using our chosen trivialization ρ(ωX)|Dx

.

Since w 6= 1, the composition

N = Nw(D×x ) → N(D×x )→ Ga

is non-zero.

Let N′ ⊂ Nw(D×x ) be such that the composition

N′ → Ga(D×x )→ Ga

is non-zero. Choose a subgroup N′′ ⊂ N0 as above. Thus, we obtain a non-trivial homomor-phism

(3.7) N′/N′′ → Ga.


3.2.5. Since N0/N′′ is unipotent, it is enough to show that the direct image with compact

supports under the map (3.6) of

(3.8) ψ|Zw,levelx/N′′ ∈ D-mod(Zw,levelx/N′′)

vanishes.

However, this follows from the fact that the object (3.8) is equivariant against the pull-backof

A-Sch ∈ D-mod(Ga)

under a non-trivial homomorphism N′/N′′ → Ga, namely, the homomorphism (3.7).

Part II: Geometry of Gm-actions on moduli problems

4. Gm-actions on classifying stacks

4.1. Attractors and repellers.

4.1.1. Let Y be an arbitrary prestack (i.e., a contravariant functor from the category of schemesto that of groupoids), equipped with an action of Gm.

We let Fixed(Y), Attr(Y) and Repel(Y) denote the prestacks given by

Hom(S,Fixed(Y)) = HomGm(S,Y),

and

Hom(S,Attr(Y)) = HomGm(S × A1,Y), Hom(S,Repel(Y)) = HomGm(S × A1,Y),

respectively, where in the case of attractors the action of Gm on A1 is the standard one, and inthe case of repellers its inverse.

4.1.2. We have the natural forgetful map Fixed(Y)→ Y, as well as the maps

Attr(Y)→ Y and Repel(Y)→ Y,

given by evaluation at 1 ∈ A1, and the maps

Fixed(Y)→ Attr(Y) and Fixed(Y)→ Repel(Y),

given by the projection A1 → pt.

The latter maps admit respective left inverses

Attr(Y)→ Fixed(Y) and Repel(Y)→ Fixed(Y),

given by evaluation at 0 ∈ A1.

4.1.3. It is shown in [Dr] that if Y is a (resp., separated) scheme of finite type, then so areFixed(Y), Attr(Y) and Repel(Y).

If Y is a scheme, on which the Gm-action is locally linear (see [DrGa3] for what this means),then the above representability is much easier to establish.

When Y is an affine scheme, the maps

Fixed(Y)→ Attr(Y)→ Y

are all closed embeddings.


4.1.4. We have the following general observation:

Proposition 4.1.5. Let N be a unipotent group, acted on by Gm with strictly positive eigevalueson the Lie algebra. Consider the resulting Gm-action on the stack pt /N . Then:

(a) Fixed(pt /N) ' pt.

(b) The forgetful map Attr(pt /N)→ pt /N is an isomorphism.

(c) The maps Fixed(pt /N)→ Repel(pt /N)→ Fixed(pt /N) are isomorphisms.

This proposition implies that we have the following commutative diagrams

pt −−−−→ Fixed(pt /N)y ypt /N −−−−→ Attr(pt /N)y y

pt −−−−→ Fixed(pt /N)

andpt −−−−→ Fixed(pt /N)y ypt −−−−→ Repel(pt /N)y ypt −−−−→ Fixed(pt /N)

with horizontal arrows being isomorphisms.

Proof. Filtering N , we can assume that N = Ga with Gm acting by the n-th power power ofthe standard character, where n > 0.

Let S be an arbitrary test-scheme, and let E be an Ga-bundle on S (resp., S×A1) is equippedwith a structure of Gm-equivariance with respect to the above Gm-action on pt /Ga (resp., andthe corresponding Gm-action on A1).

With no restriction of generality we can assume that S is affine. Then E is non-canonicallytrivial. Let V denote the Gm-module of automorphisms of E. In case (a) we have

V = Γ(S,OS),

with the standard action of Gm. In cases (b) and (c), we have

V = Γ(S,OS)⊗ k[t],

where Gm acts by the n-th power of standard character on Γ(S,OS) and where t has degree −1(resp., 1).

Note that E admits a Gm-equivariant trivialization: indeed an osbtruction would be anelement of H1(Gm, V ), whereas this group is 0.

Now, points (a) and (c) of the lemma follow from the fact that in these cases H0(Gm, V ) = 0.Point (b) follows from the fact that in this case evaluation at 1 ∈ A1 defines an isomorphism

H0(Gm, V )→ Γ(S,OS).


4.2. A digression: some maps related to parabolic subgroups. This subsection willcontain some tautological manipulations and notation, which will be used in Sect. 4.3.

4.2.1. In what follows we will use the following observation:

For a parabolic P with Levi quotient M there exists a canonically defined map1

(4.1) pt /M → pt /P,

corresponding to split P -bundles, right inverse to the tautological projection pt /P → pt /M .

Indeed, since the different splittings of the projection P → M are uniquely conjugate bymeans of the of the unipotent radical of P , they give rise to canonically isomorphic splittingsof the map pt /P → pt /M .

4.2.2. Let γ : Gm → G be a homomorphism. Let Mγ be the centralizer of γ, which is the sameas Fixed(G), where Gm acts on G by conjugation via γ. Let Pγ be the subgroup Attr(G).

It is easy to see that Pγ is a parabolic subgroup, and the maps

Mγ → Pγ →Mγ

realize Mγ as both the Levi subgroup and the Levi quotient group of Pγ .

4.2.3. Note that the homomorphism γ : Gm → Z(Mγ) defines an action of Gm on the identityautomorphism of pt /Mγ . Hence, we obtain a map

(4.2) pt /Mγ → Maps(pt /Gm,pt /Mγ),

where for two prestacks Y1 and Y2 we let Maps(Y1,Y2) denote the prestack

Hom(S,Maps(Y1,Y2)) = Hom(S × Y1,Y2).

Consider the fiber product

Maps(pt /Gm,pt /Pγ) ×Maps(pt /Gm,pt /Mγ)

pt /Mγ ,

where pt /Mγ → Maps(pt /Gm,pt /Mγ) is the map (4.2).

Lemma 4.2.4. The map

Maps(pt /Gm,pt /Pγ) ×Maps(pt /Gm,pt /Mγ)

pt /Mγ →

→ Maps(pt /Gm,pt /Mγ) ×Maps(pt /Gm,pt /Mγ)

pt /Mγ = pt /Mγ

is an isomorphism.

Proof. Follows from Proposition 4.1.5(a).

1We learned this from J. Lurie.


4.2.5. Consider now the fiber product

Maps(A1/Gm,pt /Pγ) ×Maps(A1/Gm,pt /Mγ)

pt /Mγ ,

where pt /Mγ → Maps(A1/Gm,pt /Mγ) is the map

pt /Mγ(4.2)−→ Maps(pt /Gm,pt /Mγ)→ Maps(A1/Gm,pt /Mγ).

Evaluation at 1 ∈ A1 defines a map

(4.3) Maps(A1/Gm,pt /Pγ) ×Maps(pt /Gm,pt /Mγ)

pt /Mγ → pt /Pγ .

Lemma 4.2.6. The map (4.3) is an isomorphism.

Proof. Follows from Proposition 4.1.5(b).

4.2.7. From Lemma 4.2.4 we obtain a map

(4.4) pt /Mγ → Maps(pt /Gm,pt /Pγ).

From Lemma 4.2.6 we obtain a map

(4.5) pt /Pγ → Maps(A1/Gm,pt /Pγ).

Moreover, we have a commutative diagram

(4.6)

pt /Mγ(4.4)−−−−→ Maps(pt /Gm,pt /Pγ)y y

pt /Pγ(4.5)−−−−→ Maps(A1/Gm,pt /Pγ)y y

pt /Mγ(4.4)−−−−→ Maps(pt /Gm,pt /Pγ),

where the upper left vertical map is given by (4.1).

4.3. Attractors on the classifying stack.

4.3.1. Consider the stack pt /G as endowed with the trivial action of Gm. In this subsectionwe will describe the prestacks

Fixed(pt /G) and Attr(pt /G).

Note that we have

Fixed(pt /G) = Maps(pt /Gm,pt /G) and Attr(pt /G) = Maps(A1/Gm,pt /G).


4.3.2. Composing the maps in the diagram (4.6) with the map

pt /Pγ → pt /G,

from (4.2) we obtain a commutative diagram

(4.7)

pt /Mγ −−−−→ Maps(pt /Gm,pt /G)y ypt /Pγ −−−−→ Maps(A1/Gm,pt /G)y ypt /Mγ −−−−→ Maps(pt /Gm,pt /G).

We have:

Theorem 4.3.3 (V. Drinfeld). The diagram (4.7) gives rise to isomorphisms⊔γ

pt /Mγ∼−−−−→ Maps(pt /Gm,pt /G)y y⊔

γpt /Pγ

∼−−−−→ Maps(A1/Gm,pt /G)y y⊔γ

pt /Mγ∼−−−−→ Maps(pt /Gm,pt /G),

where the disjoint unions are taken with respect to the set of conjugacy classes of homomor-phisms γ : Gm → G.

The rest of this subsection is devoted to the proof of this theorem. 2

4.3.4. The asserion that

(4.8)⊔γ

pt /Mγ → Maps(pt /Gm,pt /G)

is an isomorphism is equivalent to the following two maps being isomorphisms:

(4.9) Mapsgroups(Gm, G)/AdG → Maps(pt /Gm,pt /G).

and

(4.10)⊔γ

pt /Mγ → Mapsgroups(Gm, G)/AdG .

The fact that the map (4.9) is an isomorphsm is tautological: any G-bundle on a test-schemeS is etale-locally trivial, and the obstruction to this trivialization being Gm-equivariant is givenby a map of group-schemes S ×Gm → S ×G over S.

The fact that (4.10) is an isomorphism is equivalent to the next assertion: for a homomor-phism γ : Gm → G, the resulting map from G/Mγ to Mapsgroups(Gm, G) is an open embedding.To prove this we argue as follows.

2The proof is a minor modification of the one communicated to us by V. Drinfeld.


Note that Mapsgroups(Gm, G) is an ind-scheme of ind-finite type. Write it as a union ofG-invariant closed subschemes of finite type, denote them Yi. Note that the map

G/Mγ → Mapsgroups(Gm, G)

induces an isomorphism of tangent spaces at each point. Hence, for every indiex i, for whichthis map factors as

G/Mγ → Yi → Mapsgroups(Gm, G),

the resulting map G/Mγ → Yi is an open embedding, as desired.

4.3.5. The proof that ⊔γ

pt /Pγ → Maps(A1/Gm,pt /G)

is an isomorphism relies on the following statement:

Proposition 4.3.6. For an affine test-scheme S, any Gm-equivariant G-bundle on S × A1 is(non-canonically) isomorphic to one pulled back under the map S × A1 → S.

Assuming this proposition, we finish the proof as follows:

Knowing that (4.8) is an isomorphism, we need to show that for a test-scheme S and thetrivial G-bundle on S×A1, endowed with a structure of Gm-equivariance given by γ : Gm → G,the group of its Gm-equivariant automorphisms is isomorphic to Hom(S, Pγ). However, thegroup in question is

HomGm(S × A1, G),

and the assertion follows from the fact that Pγ := Attr(G).

4.3.7. Proof of Proposition 4.3.6. Let S = Spec(A) and A1 = Spec(k[t]). First, we note thatthe Beauville-Laszlo theorem implies that we can replace S×A1 by Spec(A[[t]]). Further, sincewe are dealing with G-bundles, we can replace Spec(A[[t]]) by the formal scheme

Spf(A[[t]]) = colimi

Spec(A[t]/ti).

Now, the required assertion follows by induction on i by standard deformation theory.

4.4. Fixed points on the affine Grassmannian. The contents of this subsection are notneeded for the rest of the paper.

4.4.1. Recall that the affine Grassmannian GrG is the indscheme whose S-points are pairsconsisting of a G-bundle on S × A1 and its trivialization on S × (A1 − 0). Note that theGm-action on A1 defines a Gm on GrG. This is the so-called action by “loop rotation”.

In this subsection we will use Theorem 4.3.3 to describe the loci Fixed(GrG) and Attr(GrG).


4.4.2. Note that we have a canonical identification

Fixed(GrG) ' Maps(A1/Gm,pt /G) ×pt /G

pt,

where the map Maps(A1/Gm,pt /G)→ pt /G is given by restriction to (A1 − 0)/Gm ' pt.

Thus, from Theorem 4.3.3, we recover the following assertion from [BD]:

Corollary 4.4.3. There exists a canonical isomorphism⊔γ

G/Pγ → Fixed(GrG),

where the disjoint unions are taken with respect to the set of conjugacy classes of homomor-phisms γ : Gm → G.

4.4.4. Unwinding the definitions, we obtain that for a given γ : Gm → G, the correspondingmap

G/Pγ → Fixed(GrG)

is defined as follows.

It sends the point 1 ∈ G/Pγ to the point tγ ∈ GrG, where tγ is induced by means of γ fromthe point of

t ∈ GrGm(k),

corresponding to the trivial line bundle on A1 with the trivialization over A1 − 0 given by thecoordinate function t.

The entire map G/Pγ → Fixed(GrG) is extended by G-equivariance, where G acts on GrGvia the embedding G → G[[t]].

4.4.5. Let GrγG ⊂ GrG be the G[[t]] oribit of the point tγ . We will now rederive the followingresult of [BD]:

Corollary 4.4.6. For every γ we have a canonical identification

GrγG = Attr(GrG) ×Fixed(GrG)

G/Pγ

as sub-prestacks of GrG.

Proof. Note that the forgetful map Attr(G[[t]]) → G[[t]] is an isomorphism. Hence, we obtainthat the G[[t]]-action on GrG lifts to a G[[t]]-action on the functor Attr(GrG). In particular, weobtain that each

Attr(GrG) ×Fixed(GrG)

G/Pγ

is G[[t]]-invariant.

Since tγ ∈ Attr(GrG) ×Fixed(GrG)

G/Pγ , we obtain an inclusion

(4.11) GrγG → Attr(GrG) ×Fixed(GrG)

G/Pγ .

Since every k-point of GrG belongs to some (in fact, unique) GrγG, we obtain that (4.11)induces a bijection on k-points.

By [DrGa3], the functor Attr(GrG) ×Fixed(GrG)

G/Pγ is representable by an ind-scheme. Hence,

in order to show that (4.11) is an isomorphism, it suffices to show that the map (4.11) inducesan isomorphism of tangent spaces at tγ . The latter is straightforward.


5. Attractors and repellers on BunN,PT

5.1. Gm-action on BunN,PT .

5.1.1. Fix a regular dominant cocharacter γ : Gm → T . This defines an action of Gm on themap

PT : pt→ BunT ,

and hence on the stack

BunN,PT = BunB ×BunT

pt .

Consider the corresponding presatcks

Fixed(BunN,PT

),Attr

(BunN,PT

)and Repel

(BunN,PT

).

5.1.2. The goal of this section is to prove the following result:

Theorem 5.1.3. The diagram

(5.1)

Xλ sλ−−−−→ ZλPTπλ−−−−→ Xλ

ιλ

y y′p−Xλ ×

BunTBunB

λ−−−−→ BunN,PT

id×qyXλ,

identifies canonically with the diagram

Fixed(BunN,PT

)−−−−→ Repel

(BunN,PT

)−−−−→ Fixed

(BunN,PT

)y yAttr

(BunN,PT

)−−−−→ BunN,PTy

Fixed(BunN,PT

).

Theorem 5.1.3 is the combination of Propositions 5.2.4, 5.3.2 and Theorem 5.4.5, proved inthis bulk of this section.

Let us show how Theorem 5.1.3 implies Theorem 2.2.3.

Proof of Theorem 2.2.3. As was established in Sect. 2.4.8, it is enough to prove Theorem 2.2.3

under the assumption that Bun≤λN,PT is an algebraic space.

In this case the fact that (2.3) is an isomorphism follows from Braden’s theorem via The-orem 5.1.3. Namely, the version of Braden’s theorem for algebraic spaces (see [DrGa3]) says


that for an algebraic space Y equipped with an action of Gm, in the diagram

(5.2)

Fixed(Y)ι−−−−−→ Repel(Y)

q−−−−−→ Fixed(Y)

ι+

y yp−

Attr(Y)p+

−−−−→ Y

q

yFixed(Y)

the resulting natural transformation

(q+)∗ (p+)! → (q−)! (p−)∗

is an isomorphism when evaluated on Gm-monodromic objects.

In particular, if the action of Gm on the terms of (5.2) is obtained from an action of somegroup H via a homomorphism Gm → H, then the corresponding natural transfomation offunctors

D-mod(Y/H)⇒ D-mod(Fixed(Y)/H)

is an isomorphism.

5.2. The fixed-point locus on BunN,PT .

5.2.1. The Gm-action on BunN,PT preserves the locally closed substacks Bun=λ

N,PT . Consideralso the corresponding prestacks

Fixed(

Bun=λ

N,PT

),Attr

(Bun

=λ

N,PT

)and Repel

(Bun

=λ

N,PT

).

Consider the map, denoted ιλ,

(5.3) Xλ ' Xλ ×BunT

BunT → Xλ ×BunT

BunB ,

where the map BunT → BunB is induced by (4.1).

It is easy to see that the resulting map

Xλ ιλ−→ Xλ ×BunT

BunB ' Bun=λ

N,PT

naturally factors as

(5.4) Xλ → Fixed(

Bun=λ

N,PT

).

We claim:

Lemma 5.2.2.

(a) The map (5.4) is an isomorphism.

(b) The map Attr(

Bun=λ

N,PT

)→ Bun

=λ

N,PT is an isomorphism.

Proof. Follows from points (a) and (b) of Proposition 4.1.5, respectively.


5.2.3. Consider now the map

(5.5)⊔

λ∈Λpos

Xλ '⊔

λ∈Λpos

Fixed(

Bun=λ

N,PT

)→ Fixed

(BunN,PT

),

where the first arrow is the isomorphism of Lemma 5.2.2(a).

We are going to prove:

Proposition 5.2.4. The maps in (5.5) are isomorphisms.

Remark 5.2.5. The decomposition into locally closed Gm-invariant substacks

BunN,PT '⋃

λ∈Λpos

Bun=λ

N,PT

defines the corresponding decomposition of Fixed(BunN,PT

)as a union of locally closed sub-

prestacks

Fixed(BunN,PT

)'

⋃λ∈Λpos

Fixed(

Bun=λ

N,PT

).

So, the assertion of Proposition 5.2.4 is equivalent to the fact that the above locally closedsub-prestacks are disjoint, i.e., that each is open.

Remark 5.2.6. Assume for a moment that the pair (PT , λ) is such that Bun≤λN,PT is a separated

algebraic space. In this case the assertion that the decomposition into locally closed subsets

Fixed(

Bun≤λN,PT

)'

⋃0≤µ≤λ

Xµ

is in fact the union of connected components is immediate: indeed each Xµ is a proper scheme,and hence must be closed as a sub-algebraic space.

5.2.7. Proof of Proposition 5.2.4. We will explictly construct an inverse map to⊔λ∈Λpos

Fixed(

Bun=λ

N,PT

)→ Fixed

(BunN,PT

).

For a test-scheme S, let us be given a Gm-equivariant map S → BunN,PT , i.e., Gm-equivariant a G-bundle PG on S × X, equipped with a generalized reduction to B, with thecorresponding T -bundle being PT |S×X , on which the structure of Gm-equivariance is definedby γ by means of (4.2).

Let U ⊂ S ×X be the open subset, dense in every fiber over S, over which our reduction toB is genuine. Note that by Lemma 4.2.4, this reduction to B comes from a canonical reductionβ to T . Furthermore, this reduction is Gm-equivariant, where the structure of Gm-equivarianceon the corresponding T -bundle is defined by γ.

We need to show that PG admits a reduction β′ to a T -bundle P′T on all of S×X, such thatβ′ coincides with β when restricted to U .

Note that by Theorem 4.3.3, there exists a homomorphism γ′ : Gm → G (uniquely defined upto conjugacy) and a reduction β′′ of PG to an Mγ′ -bundle, on which the structure of equivarianceis given by γ′ by means of (4.2).

Comparing the two pieces of data over U , the injectivity of the map in Theorem 4.3.3 impliesthat γ′ = γ, Mγ′ = T and β = β′′. We set β′ := β′′, thereby finishing the proof.


5.3. Attractors on BunN,PT .

5.3.1. For λ ∈ Λpos let

Attr(BunN,PT , X

λ)

denote the preimage of

Xλ ⊂ Fixed(BunN,PT

)under the canonical projection

Attr(BunN,PT

)→ Fixed

(BunN,PT

).

We have canonically defined maps

(5.6) Xλ ×BunT

BunB → Attr(

Bun=λ

N,PT

)→ Attr

(BunN,PT , X

λ),

where the first arrow is the isomorphism of Lemma 5.2.2(b).

We claim:

Proposition 5.3.2. The maps in (5.6) are isomorphisms.

Note that Propositions 5.2.4 and 5.3.2 imply that there exists a commutative diagram

(5.7)

⊔λ∈Λpos

Xλ ∼−−−−→⊔

λ∈Λpos

Fixed(

Bun=λ

N,PT

)∼−−−−→ Fixed

(BunN,PT

)y y y⊔

λ∈Λpos

Xλ ×BunT

BunB∼−−−−→

⊔λ∈Λpos

Attr(

Bun=λ

N,PT

)∼−−−−→ Attr

(BunN,PT

)id×q

y y y⊔λ∈Λpos

Xλ ∼−−−−→⊔

λ∈Λpos

Fixed(

Bun=λ

N,PT

)∼−−−−→ Fixed

(BunN,PT

)with the horizontal arrows being isomorphisms.

Remark 5.3.3. It is not difficult to show directly that the map (5.6) is an isomorphism at thelevel of k-points, and further that it induces an isomorphism at the level of the correspondingreduced prestacks. This would suffice for the applications in this paper, since we will be dealingwith D-modules.

5.3.4. Proof of Proposition 5.3.2. We will construct an inverse to the map⊔λ∈Λpos

Attr(

Bun=λ

N,PT

)→ Attr

(BunN,PT

).

For a test-scheme S, let us be given a Gm-equivariant map S × A1 → BunN,PT , i.e., Gm-equivariant a G-bundle PG on S × X × A1, equipped with a generalized reduction α to B,with the corresponding T -bundle being PT |S×X , on which the structure of Gm-equivariance isdefined by γ by means of (4.2).

Let U ⊂ S × X × A1 be the open subset, dense in every fiber over S × A1 over which ourreduction toB is genuine. Clearly, the open subset U is Gm-invariant and α|U is Gm-equivariant.

We need to show that PG admits a reduction α′ to a B-bundle P′B on all of S × X × A1,such that α′ coincides with α when restricted to U .


Note that by Theorem 4.3.3, there exists a homomorphism γ′ : Gm → G (uniquely definedup to conjugacy) and a reduction α′′ of PG to an Pγ′ -bundle PPγ′ , such that the structure of

equivarience on the induced Mγ′ -bundle is given by γ′ by means of (4.2).

Comparing the two pieces of data over U , the injectivity of the map in Theorem 4.3.3 impliesthat γ′ = γ, Mγ′ = T and α = α′′. We set α′ := α′′, finishing the proof.

5.4. Repellers on BunN,PT .

5.4.1. Recall the Zastava space ZλPT . The action of Gm on BunN,PT is compatible with theprojection

BunN,PT → BunG,

where Gm acts trivially on BunG. Hence, it induces an action on the Zastava space ZλPT .

The following is [BFGM, ???]:

Lemma 5.4.2. The forgetful map Repel(ZλPT ) → ZλPT is an isomorphism, and we have acommutative diagram

Fixed(ZλPT )∼−−−−→ Xλy ysλ

Repel(ZλPT )∼−−−−→ ZλPTy yπλ

Fixed(ZλPT )∼−−−−→ Xλ.

5.4.3. Consider the prestack Repel(BunN,PT

). For a given λ ∈ Λpos let

Repel(BunN,PT , X

λ)

denote the preimage of the connected component

Xλ ⊂ Fixed(BunN,PT , X

λ)

under the canonical projection

Repel(BunN,PT , X

λ)→ Fixed

(BunN,PT , X

λ).

5.4.4. From Lemma 5.4.2 we obtain that the map

′p− : ZλPT → Bun≤λN,PT

gives rise to a map

(5.8) ZλPT → Repel(

Bun≤λN,PT , X

λ)


Xλ id−−−−→ Xλysλ

yZλPT

′p−−−−−→ Repel(

Bun≤λN,PT , X

λ)

yπλ yXλ id−−−−→ Xλ

commute.


We claim:

Theorem 5.4.5. The map (5.8) is an isomorphism.

5.4.6. Proof of Theorem 5.4.5. We will construct a map

Repel(BunN,PT , X

λ)→ Repel(ZλPT ) ' ZλPT ,

inverse to (5.8).

Let us be given an S-point of Repel(

Bun≤λN,PT , X

λ)

. That is, we have a Gm-equivariant

G-bundle PG on S ×X × A1 equipped with a generalized reduction to B, such that the corre-sponding T -bundle is the pullback of PT under S ×X × A1 → X.

The projection

Repel(

Bun≤λN,PT , X

λ)→ Xλ

defines a map D : S → Xλ. Consider the open subset

U := (S ×X −GraphD)× A1 ⊂ S ×X × A1,

and let us restrict our data to it.

First, we claim that the resulting generalized B-reduction is non-degenerate, when restrictedto U . This follows from the fact that it is non-degenerate when restricted to

U ×A10 ⊂ U,

and the Gm-equivariance.

Now, Proposition 4.1.5(c) implies that our data over U admits a unique isomorphism withone induced from the T -bundle PT |U . In particular, our G-bundle, when restricted to U admitsa canonical Gm-equivariant reduction α to B−, such that the induced T -bundle is PT |U .

Consider the T -bundle P′T := PT (D) on S × X. We claim that the above B−-reductionα extends to a B−-reduction over the entire S × X × A1 with the induced T -bundle beingP′T |S×X×A1 . Such an extension is automatically unique.

The existence of such an extension, provides the required map

S → Repel(ZλPT ),

and it is clear from the construction that the resulting map

Repel(

Bun≤λN,PT , X

λ)→ Repel(ZλPT ) ' ZλPT

is the inverse of (5.8).

5.4.7. The data of a Gm-equivariant bundle PG on S ×X × A1 amounts to a map

S ×X × A1/Gm → pt /G.

By Theorem 4.3.3, there exists a homomorphism γ′ : Gm → G (uniquely defined up toconjugacy) and a reduction α′ of PG to a Pγ′ -bundle PPγ′ such that on the induced Mγ′ -bundle

PMγ′ the structure of Gm-equivariance is given by γ′ by means of (4.2). Furthermore, the pair

(γ′,PMγ′ ) is determined by the restriction of PG to

S × A1 × 0 ⊂ S × A1 × A1.


Hence, γ′ = γ, M = T and PMγ′ = P′T . Furthermore, Pγ′ = B− (the effect of replacing thestandard Gm-action by its inverse swaps the attractor and the repeller for the Gm-action on G,given by means of the conjugation by γ).

Comparing the data over U , the injectivity of the map in Theorem 4.3.3 implies that α′ = α,as required.

5.5. The enhanced flag variety. This subsection is not necessary for the rest of the paper.We will explain an alternative (and in a sense, more direct) way to prove Propositions 5.2.4,5.3.2 and Theorem 5.4.5, and thus Theorem 5.1.3.

In order to slightly simplify the exposition, we will assume that G is such that [G,G] issimply connected.

5.5.1. Consider the enhanced flag variety G/N ; it is known to be quasi-affine, and we let G/Ndenote its affine closure.

We regard G/N as equipped with a G-action on the left and a T -action on the right.

Recall that the assumption that [G,G] be simply connected implies 3 that BunN,PT is definedas the open substack of

Maps(X,G\(G/N)/T ) ×Maps(X,pt /T )

pt,

where the map pt→ Maps(X,pt /T ) is given by PT .

The corresponding open substack

Maps(X,G\(G/N)/T ) ⊂ Maps(X,G\(G/N)/T )

is given by the followowing condition:

For a test-scheme S and a map S ×X → G\(G/N)/T , we require that for every geometricpoint s ∈ S, the corresponding map

X → G\G/N/T,

is such that a non-empty open subset of X hits the open substack G\(G/N)/T ⊂ G\(G/N)/T .

5.5.2. Fix a Cartan subgroup T ⊂ B ⊂ G. We will identify the set of cocharacters γ up toconjugation with the set of dominant cocharacters of T .

Let T denote the affine variety Spec(Λ+). We have an open embedding T → T , and anaction of T on T that extends the natural T -action on itself. The quotient stack

T\Tidentifies with Π

iA1/Gm, where the product is taken over the set of vertices of the Dynkin

diagram of G.

Consider the stack Maps(X,T\T ); it splits into connected components

Maps(X,T\T ) '⊔λ

Maps(X,T\T )λ,

accprding to the degree of the composed map X → pt /T .

3For a general G, the definition of BunN,PT is slightly different, see [Sch].


For each λ, let

Maps(X,T\T )λ denote the open substack of Maps(X,T\T )λ singled out bythe following condition:

For a test-scheme S and a map S × X → T\T , we require that for every geometric points ∈ S, the corresponding map

X → T\T ,is such that a non-empty open subset of X hits the open substack pt = T/T ⊂ T\T .

Note that

Maps(X,T\T )λ is empty unless λ ∈ Λpos, and in the latter case we have a canonicalidentification

Xλ '

Maps(X,T\T )λ.

The closed embedding T → G/N extends to a closed embedding

T → G/N.

In fact, T identifies with the set of T -fixed points in G/N with respect to the action givenby the diagonal embedding

T 7→ T × T → G× T.

5.5.3. Let γ be a fixed dominant regular cocharacter Gm → T . Consider the resulting actionof Gm on the stack G\G/N .

According to Theorem 4.3.3, we have

Fixed(G\(G/N)) '⊔γ′

Fixed(G\(G/N)) ×Fixed(pt /G)

pt /Mγ′ ,

where each Fixed(G\(G/N)) ×Fixed(pt /G)

pt /Mγ′ identifies with the stack

Mγ′\Fixedγ′,γ(G/N),

where the Fixedγ′,γ(G/N) denotes the fixed-point locus of the Gm-action on G/N given by

Gmγ′,γ−→ T × T → G× T.

The following is easy to see from the definitions:

Lemma 5.5.4. The only connected components of Fixed(G\(G/N)) that have a non-emptyintersection with Fixed(G\(G/N)) are those with γ′ = γ. Furthermore, the inclusion

T → Fixedγ,γ(G/N)

is an isomorphism.

5.5.5. We note that Lemma 5.5.4 readily implies Proposition 5.2.4. We shall now deduce Propo-sition 5.3.2 and Theorem 5.4.5.

Note that the stacks Bun=λ

N,PT and ZλPT identify with the stacks

Mapsλ(X,B\T/T ) ×

Maps(X,pt /T )pt and

Mapsλ(X,B−\(G/N)/T ) ×

Maps(X,pt /T )pt,

respectively, where

•

Maps(X,−) ⊂ Maps(X,−) is the usual open condition that says that a non-empty open

subset of X must hit B\T/T ⊂ B\T/T (resp., B−\(G/N)/T ⊂ B−\(G/N)/T ).


• The superscript λ in

Mapsλ indicates that we are taking the union of those connectedcomponents for which the resulting composed maps

X → B\T/T → pt /B → pt /T and X → B−\(G/N)/T → pt /B− → pt /T

are of degree λ.

Thus, in order to prove Proposition 5.3.2 and Theorem 5.4.5, it suffices to show that themaps

B\T → Attr(G\(G/N)) ×Fixed(G\(G/N))

T\T

andB−\(G/N)→ Repel(G\(G/N)) ×

Fixed(G\(G/N))

T\T

are isomorphisms.

5.5.6. Using Theorem 4.3.3 and Lemma 5.5.4 we identify the relevant connected componentsof Attr(G\(G/N)) and Repel(G\(G/N)) with

B\Attr(G/N) and B−\Repel(G/N),

where the Gm-action on G/N is given by

Gmγ,γ−→ T × T → G× T.

Thus, it is sufficient to show that for the above Gm-action on G/N , the map

T → Attr(G/N) and Repel(G/N)→ G/N

are isomorphisms, both of which assertions are obvious.

Part III: Whittaker categories

6. Spaces of rational reductions

6.1. The framework. In this subsection G will be an arbitrary algebraic group.

6.1.1. Let Y a separated scheme equipped with an action of G. Let Maps(X,G\Y ) denote theprestack classifying maps X → G\Y . In particular, for Y = pt we recover BunG.

We let Mapsgen(X,G\Y ) ×Mapsgen(X,pt /G)

Maps(X,pt /G) denote the following prestack:

For a test-scheme S, the groupoid of S-points consists of pairs (PG, f), where:

• PG is a G-bundle on S ×X;

• f is a relatively generically defined section of the associated bundle PGG× Y .

By a relatively generically defined section we mean a section defined over an open subsetU ⊂ S×X, which is dense in the fiber over any k-point of S. Note that such U is schematicallydense in S ×X.

We emphasize that the open subset U is not part of the data. I.e., we identify f1 definedover U1 and f2 defined over U2 if f1|U1∩U2

= f2|U1∩U2.

Note that we have a tautological monomorphism

Maps(X,G\Y )→ Mapsgen(X,G\Y ) ×Mapsgen(X,pt /G)

Maps(X,pt /G),

corresponding to maps that are defined on all of S ×X.


6.1.2. It is not difficult to see that if Y is affine, then the forgetful map

Mapsgen(X,G\Y ) ×Mapsgen(X,pt /G)

Maps(X,pt /G)→ BunG

is ind-schematic (i.e., the base change by an affine scheme yields an ind-scheme).

If Y admits a G-equivariant quasi-projective (resp., projective) embedding, then it is shownin [Bar] that, up to fppf sheafification, Mapsgen(X,G\Y ) ×

Mapsgen(X,pt /G)Maps(X,pt /G) can

be presented as a quotient of an algebraic stack, equal to the disjoint union of algebraic stacksquasi-projective (resp., projective) over BunG, by a proper equivalence relation.

6.1.3. In most (but not all!) of our applications, we will be interested in the case when Y is ofthe form G/H, where H is a subgroup. In this case we shall denote

BunH -genG := Mapsgen(X,G\(G/H)) ×

Mapsgen(X,pt /G)Maps(X,pt /G).

In this case we can think of BunH -genG as the stack classifying G-bundles, equipped with a

generically defined reduction to H.

Note that the prestack Maps(X,G\(G/H)) identifies with BunH .

6.1.4. Consider the particular case of Y = G, i.e., the prestack Bun1 -genG . This is one of

the versions of the Beilinson-Drinfeld Grassmannian: it classifies G-bundles, equipped with ageneric trivialization.

We will sometimes writeGrG,gen := Bun1 -gen

G .

6.2. The space of generic parabolic reductions. In what follows we will let G be a reduc-tive group.

6.2.1. Let us take Y = G/P . One of our principal actors will be the corresponding prestack

BunP -genG .

6.2.2. Note that in addition to BunP -genG , one can consider the prestack BunP , defined as in

[BG1]. We have a tautologically defined map

BunP → BunP -genG ,

which is schematic, proper and surjective at the level of k-points.

6.3. Two versions of the Whittaker space.

6.3.1. Consider the subgroups N ⊂ G and N · ZG ⊂ G, and the corresponding prestacks

BunN -genG and BunN ·ZG -gen

G ,

respectively. Note that we have a natural forgetful map → between the two, and also forgetful

maps from both to BunB -genG .

We can think of BunN -genG (resp., BunN ·ZG -gen

G ) as the prestack, classifying the data of(PG, β, γ), where:

• PG is a G-bundle S ×X;• β is a reduction of PG to a B-bundle PB , defined over an open subset U ⊂ S×X, dense

in every fiber over S;• γ is the datum of:


– (in the case of BunN -genG ) a trivialization of the induced T -bundle T

B× PB ;

– (in the case of BunN ·ZG -genG ) a trivialization γi of the line bundle αi(PB) for every

simple root αi.

6.3.2. We let QG (resp., QG,G) denote a variant of BunN -genG (resp., BunN ·ZG -gen

G ), where wemodify the meaning of γ as follows:

• (in the case of BunN -genG ) an isomorphism between the T -bundle T

B×PB and the pullback

from X of the T -bundle ρ(ωX).

• (in the case of BunN ·ZG -genG ) an identification γi between the line bundle αi(PB) and

the pullback from X of the line bundle ωX , for each simple root αi.

As before, we have the forgetful maps

QG → QG,G → BunB -genG .


QG ×BunB -gen

G

BunB .

We have the tautological Cartesian squares

QG ×BunB -gen

G

BunB −−−−→ QGy yBunB −−−−→ BunB -gen

G ,

and another Cartesian square

QG ×BunB -gen

G

BunB −−−−→ GrT,ρ(ωX),geny yBunB −−−−→ BunT ,

where GrT,ρ(ωX),gen is the prestack classifying T -bundles, equipped with a generic identificationwith ρ(ωX).

Note, however, that since T is commutative, we can canonically identify

GrT,gen ' GrT,ρ(ωX),gen

via tensoring by the T -bundle ρ(ωX).

In particular, for every m-tuple λ of elements λ1, ..., λm of Λ, we have a canonically definedmap

Xn → GrT,ρ(ωX),gen, (x1, ..., xm) 7→ ρ(ωX)(Σλi · xi),and hence a map

BunB ×BunT

Xm → QG ×BunB -gen

G

BunB → QG.

Removing the diagonals, and symmetrizing along those λi’s that are pair-wise equal, weobtain the maps

Xλ → GrT,ρ(ωX),gen,


and the decomposition of GrT,ρ(ωX),gen into the strata:

GrT,ρ(ωX),gen '⋃λ

Xλ.

Hence, we also obtain a decomposition of QG into the strata:

(6.1) QG '⋃λ

BunB ×BunT

Xλ.

6.3.4. Similarly, consider

QG,G ×BunB -gen

G

BunB ,

and the Cartesian squares

QG,G ×BunB -gen

G

BunB −−−−→ QG,Gy yBunB −−−−→ BunB -gen

G .

andQG,G ×

BunB -genG

BunB −−−−→ GrT/ZG,ρ(ωX),geny yBunB −−−−→ BunT/ZG .

Thus, we obtain the decomposition of QG,G into the strata:

(6.2) QG,G '⋃λ

BunB ×BunT/ZG

Xλ,

where the λ’s are now elements of the quotient lattice Λ/ΛZG .

Note that the group T/ZG, being the Cartan of the adjoint quotient of G, is canonicallyisomorphic to the proeduct Π

iGm, where the i-the copy of Gm mapsto into T/ZG is the i-th

fundamental coweight $i.

6.4. The degenerate Whittaker space.

6.4.1. Let P ⊂ G be again a parabolic subgroup with Levi quotient M . Note that ZM iswell-defined a subgroup of the abstract Cartan T .

Consider the subgroup N · ZM ⊂ G, and the corresponding prestack

BunN ·ZM -genG .

We have the natural forgetful map BunN ·ZM -genG → BunB -gen

G . This allows to think of

BunN ·ZM -genG as the prestack classifying the data of (PG, β, γ), where:


in every fiber over S;• γ is the datum of trivialization γi of the line bundle αi(PB) for every simple root αi,

which i is a vertex of the Dynkin diagram of M .


6.4.2. We shall denote by QG,P the variant of BunN ·ZM -genG , where we modify the data γi as

follows:

Instead of being a trivialization of αi(PB), it is now an isomorphism with the pullback fromX of the line bundle ωX .

Note that for P = G we recover the prestack QG,G introduced in Sect. 6.3.2. At the other

extreme, i.e., for P = B we have QG,B ' BunB -genG .

6.4.3. Consider the composed map

QG,P → BunB -genG → BunP -gen

G ,

and denote

QP,M := QG,P ×BunP -gen

G

BunP ,

so that we have a Cartesian square

(6.3)

QP,M −−−−→ QG,Py yBunP −−−−→ BunP -gen

G .

Note however, the we also have the following Cartesian square:

(6.4)

QP,M′qP−−−−→ QM,My y

BunPqP−−−−→ BunM .

6.5. The extended Whittaker space. We now come to the definition of our principal actor–the prestack in D-modules on which we will realize the extended Whittaker category.

In this subsection we will take G to be a reductive group with a connected center.

6.5.1. Let chG denote the scheme Spec(k[Λpos]). Note that we have an open embedding

T/ZG =:chG → chG.

In fact, chG can be written as the union of strata indexed by the parabolics of G,

chG '⋃P

chM .

We have a natural action of T on chG, which extends the tautological action of T onchG =

T/ZG, and coincides with the tautological action of T on eachchM ' T/ZM .


6.5.2. Consider the G-scheme

G/NT× chG,

and the corresponding prestack

(6.5) Mapsgen(X,G\(G/NT× chG)) ×


By definition, the above prestack classifies the data of (PG, γ, β), where:


in every fiber over S;• γ is the datum of section γi of the line bundle −αi(PB) for every simple root αi, which

is a root of M ;

For every parabolic P , the locally closed embeddingchM → chG gives rise to a locally closed

embedding of prestacks

BunN ·ZM -genG → Mapsgen(X,G\(G/N

T× chG)) ×


In terms of the modular description as (PG, γ, β), the above locally closed embedding corre-sponds to those γ, for which γi is zero for i not in the Dynkin diagram of M , and is invertiblefor i in the Dynkin diagram of M .

6.5.3. We let QextG,G denote the variant of the prestack (6.5), where we modify the meaning of

γi as follows: instead of being a section of the line bundle −αi(PB), it is now a map from theline bundle αi(PB) to the pullback of ωX .

As before, we have a decomposition of QextG,G into locally closed sub-prestacks QG,P . In

addition, we have a tautological projection

QextG,G → BunB -gen

G .

7. Induced and Whittaker categories

7.1. The principal induced category. In this subsection we will introduce the most “ele-mentary” of the categories discussed in this category:

I(G,B) := D-mod(BunB -genG )N ⊂ D-mod(BunB -gen

G ).

This subcategory is defined by imposing a certain equivariance (or, rather, invariance) con-dition. The first step in the definition is to consider a version of this condition on the “goodlocus”.

7.1.1. Definition on the good locus. Let x = (x1, ..., xn) be a finite non-empty collection ofpoints of X. We define the sufunctors

(BunB -genG )goodx ⊂ BunB -gen

G

to consist of the data (PG, β), where we require that β be defined on a Zariski neighborhood ofx (i.e., on a Zariski neighborhood of each S × xi for a test-scheme S).

Recall the notation from Sect. 3.2. Denote

Dx := tk=1,...,n

Dxk .


Restriction under Dx → X defines a map

(BunB -genG )goodx → pt /B(Dx).

Denote

(BunB -genG )levelx := (BunB -gen

G )goodx ×pt /B(Dx)

pt .

The natural action of the group B0 := B(Dx) on (BunB -genG )levelx extends to an action of

B := B(D×x ). Denote also N0 := N(Dx) and N := N(D×x ), and similarly for T .

We identify

D-mod(

(BunB -genG )goodx

)= D-mod

((BunB -gen

G )levelx)

B0

via the operation of *-pullback.

The full subcategory

D-mod(

(BunB -genG )goodx

)N ⊂ D-mod

((BunB -gen

G )goodx

)is by definition

D-mod(

(BunB -genG )levelx

)N·T0 .

7.1.2. Reformulation in terms of finite-dimensional groups. Let us spell this out in more detail,while appealing only to prestacks locally of finite type:

For a given N′ ⊂ N (see notation in Sect. 3.2.3) we can choose N′′ ⊂ N0 and T′′ ⊂ T0 smallenough so that the subgroup N′′ ·T′′ is normal in N′ ·T0. Then the quotient group

N′ ·T0/N′′ ·T′′

acts on the quotient

(BunB -genG )levelN′′·T′′ := (N′′ ·T′′)\(BunB -gen

G )levelx .

We identify

D-mod(

(BunB -genG )goodx

)' D-mod

((BunB -gen

G )levelN′′·T′′)B0/N

′′·T′′

via the operation of *-pullback (which, up to a shift, is the same as the !-pullback, as the

morphism (BunB -genG )levelN′′·T′′ → (BunB -gen

G )goodx is smooth).

Set

D-mod(

(BunB -genG )goodx

)N′ := D-mod

((BunB -gen

G )levelN′′·T′′)

N′·T0/N′′·T′′ ⊂

⊂ D-mod(

(BunB -genG )levelN′′·T′′

)B0/N

′′·T′′ ' D-mod(

(BunB -genG )goodx

).

It is easy to see that the above subcategory is independent of the choice of N′′ and T′′. Notethat since N′/N′′ is unipotent, the forgetful functor

D-mod(

(BunB -genG )goodx

)N′ → D-mod

((BunB -gen

G )goodx

)is fully faithful.


Finally, we set

D-mod(

(BunB -genG )goodx

)N := lim

N′D-mod

((BunB -gen

G )goodx

)N′ '⋂

N′

D-mod(

(BunB -genG )goodx

)N′ ⊂ D-mod

((BunB -gen

G )goodx

),

where the index N′ runs through the poset of group-subschemes of N.

7.1.3. Preservation by functors. Let Y be a sub-prestack of (BunB -genG )goodx , whose preimage

in (BunB -genG )levelx is preserved by the action of N(D×x ).

Then by the same token, it makes sense to define

D-mod(Y)N ⊂ D-mod(Y).

Let f : Y′ → Y be an inlcusion of sub-prestacks as above. Then the functor f ! sendsD-mod(Y)N to D-mod(Y′)N.

If f is schematic, so that the functor f∗ : D-mod(Y′) → D-mod(Y) is defined, then thisfunctor also sends D-mod(Y′)N to D-mod(Y)N.

Further, suppose F′ ∈ D-mod(Y′) be an object such that the partially defined left adjoint f!

to f ! is defined on F′. Then if F′ ∈ D-mod(Y′)N, then f!(F′) ∈ D-mod(Y)N.

Similarly, assume that f is schematic, and let F ∈ D-mod(Y) be such that the partially definedleft adjoint f∗ to f∗ is defined on F. Then if F ∈ D-mod(Y)N, then f∗(F) ∈ D-mod(Y′)N.

If α 7→ Yα ⊂ Y is a collection of invariant sub-prestacks as above, and assume that Y(k) =∪αYα(k). The followimg easily results from the definitions:

Lemma 7.1.4. Under the above circumstances, an object of D-mod(Y) belongs to D-mod(Y)N

if and only if its !-restriction to each Yα belongs to D-mod(Yα)N.

7.1.5. Strata-wise description. Recall the inclusion

BunB → BunB -genG ,

and note that it factrs through (BunB -genG )goodx for any x.

We have the following general assertion:

Lemma 7.1.6. Let Y be a N-invariant substack of BunB. Then:

(a) Y is the preimage under the map qB : BunB → BunT of a uniquely defined sub-prestackZ ⊂ BunT ;

(b) The functors

(qB)∗ : D-mod(Z) D-mod(Y)N : (qB)∗

are mutually inverse equivalences of categories;

(c) Under the identifiaction of point (b), the functor

(qB)∗ : D-mod(Y)→ D-mod(Z)

goes over to the right adjoint to the inclusion D-mod(Y)N → D-mod(Y).


Proof. We will us the notation of Sect. 7.1.2. The assertion follows from the fact that (over agiven quasi-compact substack of BunT ), we can choose the subgroup N′ large enough so thatthe map

(N′ ·T0/N′′ ·T′′)\(BunB -gen

G )levelN′′·T′′ → BunT

is a unipotent gerbe.

Remark 7.1.7. Note that Lemmas 7.1.4 and 7.1.6 give a “hands-on” description of

D-mod(

(BunB -genG )goodx

)N ⊂ D-mod

((BunB -gen

G )goodx

).

Namely, an object of D-mod(

(BunB -genG )goodx

)belongs to D-mod

((BunB -gen

G )goodx

)N if and

only if its !-restriction to BunB belongs to the essential image of the fullu faithful functor

q∗ : D-mod(BunT )→ D-mod(BunB).

7.1.8. Independence of the choice of x. Let x ⊂ x′ be an inclusion of non-empty collection of

points. We can regard (BunB -genG )goodx′ as a sub-prestack of (BunB -gen

G )goodx .

Let Y be a sub-prestack of (BunB -genG )goodx′ . Thus, there are a priori two different meanings

of D-mod(Y)N, depending on whether we regard it with respect to x′ or x.

However, intersecting Y with BunB , and applying Lemmas 7.1.4 and 7.1.6, we obtain thatthe above two conditions on an object of D-mod(Y) to belong to D-mod(Y)N coincide.

7.1.9. We are finally ready to define the subcategory


G ).

Namely, we let it to be the full subcategory, consisting of those objects, whose restriction to

(BunB -genG )goodx belongs to D-mod

((BunB -gen

G )goodx

)N for any non-empty collection of points

x.

Remark 7.1.10. As written, the condition of belonging to the Whittaker category involveschecking a huge family of conditions: one for each collection x. However, Sect. 7.1.8 says thatit suffices to check a much smaller set of conditions:

Note that the sub-prestacks (BunB -genG )goodx for x ∈ X form a Zariski cover of BunB -gen

G .

Now, Sect. 7.1.8 implies that to check whether a given object of D-mod(BunB -genG ) belongs to

D-mod(BunB -genG )N, it is enough to check that its restriction to each (BunB -gen

G )goodx belongsto the corresponding subcategory

D-mod(

(BunB -genG )goodx

)N ⊂ D-mod

((BunB -gen

G )goodx

).

7.1.11. Let Y be a sub-prestack of BunB -genG , such that its intersection with any (BunB -gen

G )good x

has the invariance property described in Sect. 7.1.3. We shall call such sub-prestacks N-invariant.

For a N-invariant sub-prestack we define D-mod(Y)N ⊂ D-mod(Y) by the same procedure as

when Y is all of BunB -genG . The assighment Y 7→ D-mod(Y)N has properties analogous to those

described in Sects. 7.1.3-7.1.5. In particular, Lemmas 7.1.4 and 7.1.6 and Remark 7.1.7 applyto the present context.

In particular, we have a well-defined functor

(ιT )! : I(G,B)→ D-mod(BunT )



D-mod(BunB -genG )

(ιB)!

−−−−→ D-mod(BunB)x xD-mod(BunB -gen

G )N −−−−→ D-mod(BunB)N

=

x x(qB)∗

I(G,B)(ιT )!

−−−−→ D-mod(BunT )

commute, where ιB denotes the inclusion BunB → BunB -genG .

7.2. Averaging. In this subsection we will show that the tautological embedding


G ).

admits a continuous right adjoint, denoted AvN∗ ; morever, this right adjoint is “nicely behaved”.

7.2.1. Let Y be an N-invariant sub-prestack of BunB -genG . We will decsribe the functor AvN

∗right adjoint to the tautological embedding D-mod(Y)N → D-mod(Y).

We shall first to do when Y is a sub-prestack of (BunB -genG )goodx for some x.

7.2.2. Recall the notation of Sect. 7.1.2. Set

YlevelN′′·T′′ := Y ×(BunB -gen

G )goodx

(BunB -genG )levelN′′·T′′ ;

this is a B0/N′′ ·T′′-torsor over Y.

Let

D-mod(YlevelN′′ )N′·T0/N

′′·T′′ ⊂ D-mod(YlevelN′′ )B0/N′′·T′′ ' Y

be the corresponding equivariant category. This category is independent of the choice of N′′ ·T′′;we shall denote it by D-mod(Y)N

′.

The embedding of this subactegory admits a right adjoint, given by *-averaging along the(unipotent, finite-dimensional) group N′/N′′ on YlevelN′′·T′′ . We shall denote this right adjoint

by AvN′

∗ .

7.2.3. By definition

D-mod(Y)N = ∩N′

D-mod(Y)N′⊂ D-mod(Y).

A priori, the right adjoint to the inclusion D-mod(Y)N → D-mod(Y) is given by

(7.1) F 7→ limN′

AvN′

∗ (F).

Thus, our task is to show that the functor in (7.1) is continuous.


7.2.4. We claim that for a quasi-compact test-scheme S and an S-point of Y, the family offunctors that takes F ∈ D-mod(Y) to

N′ 7→ AvN′

∗ (F)|Sstabilizes.

Indeed, by since the restriction functor under BunB → BunB -genG is conservative, it is enough

to assume that our S-point factors through BunB .

Now the stabilization follows from the same argument as in the proof of Lemma 7.1.6: over aquasi-compact substack of BunB , equivariance with respect to N is equivalent to equivariancewith respect to some N′, provided that N′ ⊂ N is large enough.

7.2.5. The above stabilization property implies that for an inclusion f : Y′ → Y, we have acommutative diagram

(7.2)

D-mod(Y)f !

−−−−→ D-mod(Y′)

AvN∗

y yAvN∗

D-mod(Y)Nf !

−−−−→ D-mod(Y′)N.

In addition, if f is schematic, the following diagram is also commutative:

D-mod(Y′)f∗−−−−→ D-mod(Y)

AvN∗

y yAvN∗

D-mod(Y′)Nf∗−−−−→ D-mod(Y)N.

7.2.6. The above properties imply that the functor AvN∗ is continuous also in the context of Y

being an N-invariant sub-prestack of all of BunB -genG (rather than of (BunB -gen

G )goodx).

Furthermore, the commutativity of the diagrams in Sect. 7.2.5 continues to hold in thiscontext as well.

7.2.7. Composing the functor

(penh)! : D-mod(BunG)→ D-mod(BunB -genG )

with the functor

AvN∗ : D-mod(BunB -gen

G )→ D-mod(BunB -genG )N =: I(G,B),

we obtain a functor that we denote

CTenhB : D-mod(BunG)→ I(G,B).

We call it the enhanced principal Constant Term functor. The commutation of the diagram(7.2) and Lemma 7.1.6(c) imply:

Corollary 7.2.8. There exists a canonical isomorphism of functors D-mod(BunG) →D-mod(BunT )

(ιT )! CTenhB ' CTB ,

where we remind that CTB := (qB)∗ (pB)!, and (ιT )! is as in Sect. 7.1.11.


7.3. General parabolic induced categories. In this subsection we fix a parabolic subgroupP . We will generalize the discussion of the previous subsection, and define a certain full sub-category

I(G,P ) := D-mod(BunP -genG )N(P) ⊂ D-mod(BunP -gen

G ).

7.3.1. The definition is similar to that of I(G,B) with the following modifications:

• Level structure is taken with respect to the group P (instead of B);• Extra equivariance is imposed with respect to the group N(P) := N(P )(D×x ) (instead

of N := N(D×x );

7.3.2. Let Y ⊂ BunP -genG be a sub-prestack, such that its preimage in any (BunP -gen

G )goodx isinvariant with respect to the action of the corresponding group-indscheme N(P). We shall callsuch sub-prestacks N(P)-invariant.

By a similar token, for a N(P)-invariant sub-prestack Y we define the full subcategory

D-mod(Y)N(P) ⊂ D-mod(Y).

The assignment Y 7→ D-mod(Y)N(P) has properties similar to those described in in Sects. 7.1.3-7.1.5.

Analogously to Lemma 7.1.6, we have:

Lemma 7.3.3. Let Y be a N(P)-invariant substack of BunP . Then:

(a) Y is the preimage under the map qP : BunP → BunM of a uniquely defined sub-prestackZ ⊂ BunM ;

(b) The functors

(qP )∗ : D-mod(Z) D-mod(Y)N(P) : (qP )∗

are mutually inverse equivalences of categories;

(c) Under the identifiaction of point (b), the functor

(qP )∗ : D-mod(Y)→ D-mod(Z)

goes over to the right adjoint to the inclusion D-mod(Y)N(P) → D-mod(Y).

In particular, we have a well-defined functor

(ιM )! : I(G,P )→ D-mod(BunM )


D-mod(BunP -genG )

(ιP )!

−−−−→ D-mod(BunP )x xD-mod(BunP -gen

G )N −−−−→ D-mod(BunP )N(P)

=

x x(qP )∗

I(G,P )(ιM )!

−−−−→ D-mod(BunM )

commute, where ιP denotes the inclusion BunP → BunP -genG .

7.3.4. For any Y as in Lemma 7.3.3 we shall denote by AvN(P)∗ the right adjoint to the embedding

D-mod(Y)N(P) → D-mod(Y).

It has properties parallel to those of the functor AvN∗ in Sect. 7.2.


7.3.5. We define the enhanced parabolic Constant Term functor

CTenhP : D-mod(BunG)→ D-mod(BunP -gen

G )N(P) = I(G,P )

to be the composition

AvN(P)∗ (penh

P )!.

As in Corollary 7.2.8 we have a canonical isomorphism of functors

(7.3) (ιM )! CTenhP ' CTP : D-mod(BunG)→ D-mod(BunM ),

where CTP := (qP )∗ (pP )!.

7.4. Two versions of the (non-degenerate) Whittaker category. In this subsection we

will discuss yet another version of the category D-mod(BunB -genG )N, by imposing equivariance

against a non-degenerate character.

We will define full subcategories

Whit(G) := D-mod(QG)N,χG ⊂ D-mod(QG)

and

Whit(G,G) := D-mod(QG,G)N,χG ⊂ D-mod(QG,G).

First, we need the following additional ingredient.

7.4.1. As in Sect. 3.2, we choose a trivialization of the T -bundle ρ(ωX). Recall (see Sect. 3.2.4)that such a trivialization defines a homomorphism

(7.4) N→ Ga,

which is trivial on N0, and hence a homomorphism N′/N′′ → Ga for any

N′′ ⊂ N0 ⊂ N′ ⊂ N

as in Sect. 3.2.3. Let χG,N′/N′′ denote the corresponding character sheaf on N′/N′′, equal tothe pull-back of the A-Sch under the homomorphism (7.4).

7.4.2. Now the definition of the full subcategories

D-mod(QG)N,χG ⊂ D-mod(QG) and D-mod(QG,G)N,χG ⊂ D-mod(QG,G)

follows the procedure in Sect. 7.1, with the following modifications in Sect. 7.1.2:

• Level structure is considered with respect to the group N ;• Instead of plain equivariance with respect to the group N′ · T0/N

′′ · T′′, we requireequivariance against the character sheaf χG,N′/N′′ on the group N′/N′′.

7.4.3. Let Y be a N-invariant sub-prestack of QG (resp., QG,G). By a similar token we introducethe full subcategory

D-mod(Y)N,χG ⊂ D-mod(Y).

The assignment Y 7→ D-mod(Y)N,χG has properties similar to those described in Sect. 7.1.3,and in particular, the analog of Lemma 7.1.4 holds.

In addition, the inclusion D-mod(Y)N,χG → D-mod(Y) admits a continuous right adjoint,

denoted AvN,χG∗ , with properties parallel to those of the functor AvN

∗ in Sect. 7.2.


Note that for the forgetful map f : QG → QG,G, the functor f ! sends Whit(G,G) to Whit(G),and the diagram

(7.5)

D-mod(QG)Av

N,χG∗−−−−−→ Whit(G)

f !

x xf !

D-mod(QG,G)Av

N,χG∗−−−−−→ Whit(G,G).

7.4.4. The functor(s) of non-degenerate Whittaker coefficients. We define the functors of non-degenerated Whittaker coefficients:

W-coeffG : D-mod(BunG)→Whit(G) and W-coeffG,G : D-mod(BunG)→Whit(G,G)

to be

AvN,χG∗ (rG)! (penh

B )! and AvN,χG∗ (rG,G)! (penh

B )!,

respectively.

We remind that rG and rG,G denote the maps

QG → BunB -genG and QG,G → BunB -gen

G ,

respectively.

7.4.5. Strata-wise description. One substantial difference between the Whittaker categories

(i.e., D-mod(QG)N,χG or D-mod(QG,G)N,χG) and D-mod(BunB -genG ) is their strata-wise de-

scription, cf. Sect. 7.1.5.

Recall that the prestack QG (resp., QG,G) is the union of the strata given by (6.1) (resp.,(6.2)), i.e., of the form

(7.6) BunB ×BunT

Xλ and BunB ×

BunT/ZG

Xλ,

respectively. Let g denote the map from such a stratum to the correspondingXλ.

We have the following version of Lemma 7.1.6 (cf. [FGV]):

Lemma 7.4.6. Let Y be a N-invariant sub-prestack of one of the strata in (7.6).

(a) Y is the preimage under g of a uniquely defined sub-prestack Z ⊂Xλ.

(b) The category D-mod(Y)N,χG is zero unless λ is dominant, i.e., if λ = λ1, ..., λm, then allλj ∈ Λ+.

(c) If λ is dominant, we have a canonically defined map ev : Y→ A1, and the operation

F ∈ D-mod(Z) 7→ g∗(F)∗⊗ ev∗(A-Sch) ∈ D-mod(Y)

defines an equivalence

D-mod(Z)→ D-mod(Y)N,χG .

7.5. Degenerate Whittaker categories. Let P be again a parabolic subgroup. Considerthe prestack QG,P . In this subsection we will define a full subcategory

Whit(G,P ) := D-mod(QG,P )N,χP ⊂ D-mod(QG,P ),

which combines some of the ingredients of I(G,P ) and Whit(G,G).


7.5.1. The definition of Whit(G,P ) is similar to that of Whit(G,G), with the following modi-fications:

• Level structure is taken with respect to the group N · ZM (instead of N · ZG);• Instead of the (non-degenerate) homomorphism N → Ga used in Sect. 7.4.1, we use

the homomorphism

N → NM → Ga,

whereNM ' N/N(P ) is the unipotent radical of the Borel subgroup of the Levi quotientM , and NM → Ga is the corresponding (non-degenerate) homomorphism for M .• Equivariance is imposed with respect to subgroups of the form

N′ · (ZM)0/N′′ · (ZM)′′

for

N′′ ⊂ N0 ⊂ N′ ⊂ N and (ZM)′′ ⊂ (ZM)0

so that N′′ · (ZM)′′ is normal in N′ · (ZM)0.

Note that for P = B, we get Whit(G,B) = I(G,B), and for P = G we recover Whit(G,G).

7.5.2. Let Y be a N-invariant sub-prestack of QG,P . By a similar token we introduce the fullsubcategory

D-mod(Y)N,χP ⊂ D-mod(Y).

The assignment Y 7→ D-mod(Y)N,χP has properties similar to those described in Sect. 7.1.3,and in particular, the analog of Lemma 7.1.4 holds.

In addition, the inclusion D-mod(Y)N,χP → D-mod(Y) admits a continuous right adjoint,

denoted AvN,χP∗ , with properties parallel to those of the functor AvN

∗ in Sect. 7.2.

7.5.3. Recall Cartesian diagram (6.4):

QP,M′qP−−−−→ QM,M

rP,M

y yrM,M

BunPqP−−−−→ BunM .

As in Lemma 7.3.3, we obtain the following:

Lemma 7.5.4. Let Y be a sub-prestack of QP,M , invariant with respect to N. Then:

(a) Y is the preimage under the map ′qP of a uniquely defined sub-prestack Z ⊂ BunM invariantwith respect to NM,

(b) The functors

(′qP )∗ : D-mod(Z)NM,χM D-mod(Y)N,χP : (′qP )∗

are mutually inverse equivalences of categories.


7.5.5. We define the functor of P -degenerate Whittaker coefficients:

W-coeffG,P : D-mod(BunG)→Whit(G,P )

to beAvN,χP∗ r!G,P (penh

P )!,

where rG,P denotes the projection QG,P → BunP -genG .

In addition, we have the functor W-coeffP,P : I(G,P )→Whit(G,P ), defined as

AvN,χP∗ r!G,P .

It follows from the definitions that we have

(7.7) W-coeffG,P 'W-coeffP,P CTenhP .

7.5.6. Consider the Cartesian diagram (6.3)

QP,M′ι!P−−−−→ QG,P

rP,M

y yrG,P

BunPι!P−−−−→ BunP -gen

G .

It follows from the definitions and Lemma 7.5.4 that we have the following isomorphism offunctors

I(G,P )→Whit(M,M),

namely,

(7.8) (′qP )∗ (′ιP )! W-coeffP,P 'W-coeffM,M (ιM )!,

where we recall that (ιM )! denotes the functor I(G,P )→ D-mod(BunM ) equal to (qP )∗ (ιP )!.

In particular, combing (7.8) and (7.7), we obtain the following isomorphism of functors

D-mod(BunG)→Whit(M,M),

namely:

(7.9) (′qP )∗ (′ιP )! W-coeffG,P 'W-coeffM,M CTP .

7.5.7. Let us summarize the above discussion. We have the usual Constant Term functor

CTP : D-mod(BunG)→ D-mod(BunM ),

but this functor loses a lot of information. Its enhancement is the functor

CTenhP : D-mod(BunG)→ I(G,P ),

and the functor CTP is obtained from CTenh by restriction along ιP : BunP → BunP -genG , i.e.,

CTP ' (ιM )! CTenhP .

We can further compose the functor CTP with the functor of non-degenerate Whittakercoefficients for M , i.e., we obtain the functor

W-coeffM,M CTP : D-mod(BunG)→Whit(M,M).

This functor also loses information because CTP does. Its enhancement is the functor

W-coeffG,P : D-mod(BunG)→Whit(G,P ).

The relationship between W-coeffM,M CTP and W-coeffG,P is also given by retsriction to thestrata along the ′ιP , i.e., by the isomorphism (7.9).


7.6. The extended Whittaker category. Our final goal in this section is to define the fullcategory

Whitext(G) ⊂ D-mod(Qext).

This category will contain the categories Whit(G,P ) as strata, where P ranges through theposet of parabolics subgroups of G.

7.6.1. We shall write the definition by appealing to group-schemes and prestacks of infinite typeas in Sect. 7.1.1, leaving the reformulation in terms of objects of finite type (in the spirit ofSect. 7.1.2) to the reader.

For a fixed non-empty collection of points x, let (Qext)goodx be the corresponding sub-prestackof Qext, where we require that the data of both β and γ (see Sect. 6.5.3) be defined near x.

7.6.2. We have a canonically defined map (Qext)goodx → pt /B0. Consider the fibe product

(Qext)levelx := (Qext)goodx ×pt /B0

pt .

The action of B0 on (Qext)levelx canonically extends to an actio of B.

Note now that the data of γ in the definition of Qext gives rise to a canonically defined map

(7.10) (Qext)levelx → ΠiωDx =: chG,x.

The map (7.10) is equivariant with respect to the group T0, where the latter acts on chG,xvia the map

T → T/ZG ' ΠiGm.

The scheme chG,x identifies with the scheme of group homomorphisms N → Ga that aretrivial on N0. Thus, the group-indscheme N × chG,x over chG,x comes equipped with a ho-

momorphism, denoted χext, to the constant group-scheme Ga × chG,x. Furthemore, χext isinvariant with respect to the action of T0, which acts by conjugation on N, and in the wayspecified above on chG,x.

Thus, viewing (Qext)levelx as a prestack equipped with a N-action over chG,x by means of

the map (7.10), it makes sense to consider the equivariant category

D-mod((Qext)levelx

)N,χext

,

equipped with a fully faithful forgetful functor to D-mod((Qext)levelx

)N0 .

Furthemore, since we have a compatible system of actions of T0 on all of the above objects,we can consider the category (

D-mod((Qext)levelx

)N,χext

)T0 ,

which is equipped with a fully faithful forgetful functor to

D-mod((Qext)levelx

)N0·T0 ' D-mod

((Qext)goodx

).

We set

D-mod((Qext)goodx

)N,χext

:=(

D-mod((Qext)levelx

)N,χext

)T0 .


7.6.3. The definition of the category

Whitext(G) := D-mod(Qext)N,χext

⊂ D-mod(Qext)

is obtained from that of

D-mod((Qext)goodx

)N,χext

⊂ D-mod((Qext)goodx

)by the procedure parallel to that in Sects. 7.1.9.

In particular, the definition of the subcategory

D-mod(Y)N,χext

⊂ D-mod(Y)

makes sense for any N-invariant sub-prestack of Y ∈ Qext, and properties similar to thosedescribed in Sect. 7.1.3, and the analog of Lemma 7.1.4 hold.

Further, the inclusion D-mod(Y)N,χext

→ D-mod(Y) admits a continuous right adjoint, de-

noted AvN,χext

∗ with properties analogous to those described in Sect. 7.2.

7.6.4. Recovering the degenerate Whittaker categories. Recall that for a fixed parabolic P wehave a canonically defined locally closed embedding

iP : QG,P → Qext.

Thus, taking Y = QG,P , we obtain a full subactegory

D-mod(QG,P )N,χext

⊂ D-mod(QG,P ).

Proposition 7.6.5. We have

D-mod(QG,P )N,χext

= Whit(G,P )

as subcategories of D-mod(QG,P ).

Proof. By the construction of both categories, it is enough to identify the two subactegories of

D-mod(QgoodxG,P ), i.e.,

D-mod(QgoodxG,P )N,χP and D-mod(Q

goodxG,P )N,χ

ext

.

The two categories in question are defined as follows. For D-mod(QgoodxG,P )N,χP we consider

the prestack

QgoodxG,P ×

pt /N0·(ZM)0

pt,

and impose equivariance for the group N · (ZM)0 against the character

χP : N · (ZM)0 → Ga.

For D-mod(QgoodxG,P )N,χ

ext

we consider the prestack

QgoodxG,P ×

pt /B0

pt,

equipped with a map to chG,x, and impose equivariance with respect to N against the family

of characters χext, together with an equivariance with respect to T0.

Note, however, that the restriction of the map (7.10) to

QgoodxG,P ×

pt /B0

pt ⊂ (Qext)levelx


maps to the locally closed subschemechM,x ⊂ chG,x

(see the notation in Sect. 6.5.1).

Note also that the action of T0 onchM,x is transitive. A choice of a trivialization of ρ(ωX)

on Dx fixes a point inchM,x whose stabilizer equals (ZM)0. The preimage of this point in

QgoodxG,P ×

pt /B0

pt

identifies canonically with

QgoodxG,P ×

pt /N0·(ZM)0

pt .

These makes the identification of the two categories in question manifest.

7.6.6. The functor of extended Whittaker coefficients. Let rext denote the map

Qext → BunB -genG .

We define the functor of extended Whittaker coefficients

W-coeffextG : D-mod(BunG)→Whitext(G)

to be the composition

AvN,χext

∗ (rext)! (penhB )!.

By Sect. 7.6.4, for a fixed parabolic P we have

i!P W-coeffextG 'W-coeffG,P ,

as functors D-mod(BunG)→Whit(G,P ).

8. Existence of certain left adjoints

8.1. What we want to do.

8.1.1. In the previous section we introduced several categories

I(G,P ) ⊂ D-mod(BunP -genG ); Whit(G) ⊂ D-mod(QG); Whit(G,P ) ⊂ D-mod(QG,P )

andWhitext(G) ⊂ D-mod(Qext).

These categories came along with the naturally defined functors

(ιM )! : I(G,P )→ D-mod(BunM );

CTenhP : D-mod(BunG)→ I(G,P );

W-coeffG : D-mod(BunG)→Whit(G);

W-coeffG,G : D-mod(BunG)→Whit(G,G);

f ! : Whit(G,G)→Whit(G);

W-coeffP,P : I(G,P )→Whit(G,P );

W-coeffG,P : D-mod(BunG)→Whit(G,P );

(′qP )∗ (′ιP )! : Whit(G,P )→Whit(M,M);

i!P : Whitext(G)→Whit(G,P );


W-coeffextG : D-mod(BunG)→Whitext(G).

Our goal in this section is to show that all of the above functors admit left adjoints. Suchquestions are non-obvious because the !-pushforward is not in general defined even for schematicmorphisms, when one works with non-holonomic D-modules.

The existence of these left adjoints is important for geometric Langlands, because objectsobtained by applying these functors are compact generators of many categories of interest.Moreover, the left adjoints have clear counterparts on the spectral side.

8.1.2. The left adjoints of the functors in Sect. 8.1.1 can be split into three groups, in each ofwhich the reasons of its existence is different:

Group one is(ιM )!, CTenh

P and (′qP )∗ (′ιP )!.

Here the idea behind the existence of the left adjoint in question is Drinfeld’s compactification

BunP and a certain ULA property, see Sect. 8.2.1 below.

Group two consists of the functor f !. In this case, the reason for the existence of the leftagoint is a certain property of the prestack Mapsgen(X,Gm).

Group three consists of W-coeffG, W-coeffG,G and i!P . In this case the reason for the existenceof the left adjoint in question is the Hecke action.

8.1.3. The existence of the left adjoint to W-coeffP,P , to be denoted PoincP,P , is a formalcorollary of the existence of the other functors, specifically, W-coeffG,G, ι!M and (′qP )∗ (′ιP )!.

Indeed, since the functor (′qP )∗ (′ιP )! is conservative and admits a left adjoint, in order toprove the existence of the left adjoint to W-coeffP,P , it suffices to prove the existence of a leftadjoint of the composed functor

(′qP )∗ (′ιP )! W-coeffP,P .

Now, using (7.8), the result follows from the existence of the left adjoints to ι!M andW-coeffM,M (so we are using the existence of the left adjoint to W-coeffG,G for the group M).

Note that the existence of the left adjoints of W-coeffP,P and of CTenhP , respectively, implies

the existence of the left adjoint of W-coeffG,P , to be denoted PoincG,P , since

W-coeffG,P = W-coeffP,P CTenhP .

8.1.4. Finally, we note that the existence of the left adjoint of i!P and W-coeffG,P implies the

existence of a left adjoint of W-coeffextG ; the latter willbe deonoted by Poincext

G .

8.2. Using Drinfeld’s compactification.

8.2.1. Recall the map

kP : BunP → BunP -genG ,

see Sect. 6.2.2. Let ιP denote the open embedding

BunP → BunP ;

we have kP ιP = ιP .

Let qP denote the map

BunP → BunM .

We will use the following key property of ths object

(ιP )!(kBunP ) ∈ D-mod(BunP ).


Namely, it is established in [BG1] that (ιP )!(kBunP ). is ULA with respect to the map q.

8.2.2. Let us show that the functor

(ιM )! : I(G,P )→ D-mod(BunM )

admits a left adjoint, to be denoted (ιM )!.

Since the map kP is schematic and proper, the functor

(kP )! : D-mod(BunP -genG )→ D-mod(BunP ),

left adjoint to (kP )!, is defined. Hence, to prove the existence of the left adjoint to

(ιM )! = (q)∗ (ιP )!,

it suffices to show that the left adjoint to

(q)∗ (ιP )! : D-mod(BunP )→ D-mod(BunM )

exists.

Now, the ULA property of (ιP )!(kBunP ) immediately implies that the functor

(ιP )! (qP )∗ : D-mod(BunM )→ D-mod(BunP )

is defined. In fact

(ιP )! (qP )∗(F) ' (ιP )!(kBunP )!⊗ (qP )!(F)[2 dim(BunM )].

8.2.3. We will now show that the functor

(′qP )∗ (′ιP )! : Whit(G,P )→Whit(M,M)

admits a left adjoint, to be denoted (′ιP )! (′qP )∗.

We will show more generally that the functor

(′qP )∗ (′ιP )! : D-mod(QG,P )→ D-mod(QM,M )

admits a left adjoint.

Note that the morphism

BunP ×BunM

QM,M → QP,M → QG,P

extends to a morphism

kQP : BunP ×

BunMQM,M → QG,P .

Moreover, the morphism kQP is schematic and proper. Hence, it is enough to show that the

functor

D-mod(BunP ×BunM

QM,M )(ιP×idQM,M

)!

−→ D-mod(BunP ×BunM

QM,M )(qP×idQM,M

)∗−→ D-mod(QM,M )

admits a left adjoint, i.e., that the functor

(ιP × idQM,M )! (qP × idQM,M )∗

is defined.

The existence of the latter functor follows again from the fact that

(ιP )!(kBunP ) ∈ D-mod(BunP )


is ULA with respect to the map q. Indeed, the left adjoint in question is isomorphic to

F 7→ (idBunP

×rM,M )! (ιP )!(kBunP )!⊗ (qP × idQM,M )!(F)[2 dim(BunM )].

8.2.4. We will now show that the functor

CTenhP : D-mod(BunG)→ I(G,P )

admits a left adjoint, to be denoted EisenhP .

We shall give two (closely related) proofs.

The first proof uses the well-definedness of the functor (ιM )!. Since the functor (ιM )! isconservative and admits a left adjoint, it suffices to show that the composed functor

(ιM )! CTenhP : D-mod(BunG)→ D-mod(BunM )

admits a left adjoint.

Recall that (ιM )! CTenhP ' CTP (see (7.3)), and the latter functor does admit a left adjoint:

EisP := (pP )! (qP )∗,

see [DrGa4].

Note, however, that the existence of EisP also used the stack BunP and the ULA property

of (ιP )!(kBunP ) ∈ D-mod(BunP ).

8.2.5. The second proof of the existence of EisenhP is in a sense more streamlined. We will show

that the functor

(penhP )! : D-mod(BunG)→ D-mod(BunP -gen

G )

already admits a left adjoint, to be denoted (penhP )!.

Since the morphism kP : BunP → BunP -genG is surjective on field-valued points, the functor

(kP )! is conservative. Hence, it is enough to show that the composition

(kP )! (penhP )! = (pP )!

admits a left adjoint, where

pP : BunP → BunG .

The latter is, however, immediate since the morphism pP is schematic and proper.

8.3. Rational maps into a torus.

8.3.1. For a connected algebraic group H consider the group-prestack

Mapsgen(X,H)

and the prestack

GrH,gen := BunH ×Mapsgen(X,pt /H)

pt .

We have have a natural projection

(8.1) GrH,gen → BunH .

For any test-scheme S and an S-point of BunH , the set of its left to an S-point of GrH,gen isa torsor over the group of S-points of Mapsgen(X,H). Furthermore, according to [DS], thistorsor is non-empty locally in the Zariski topology on S.

The following is the main result of [Ga] (with (b) being an easy corollary of (a)):


Theorem 8.3.2.

(a) The functor Vect→ D-mod(Mapsgen(X,H)) is fully faithful.

(b) The !-pullback functor along (8.1)

D-mod(BunH)→ D-mod(GrH,gen).

is fully faithful.

8.3.3. We will need the following two additional properties of the prestack GrH,gen:

Lemma 8.3.4.

(a) GrH,gen can be written as a colimit of prestacks, each representable by a scheme of finitetype, with transition maps being proper

(b) Assume that H is reductive. Then GrH,gen can be written as a colimit of prestacks, eachrepresentable by a projective scheme.

8.3.5. Assume now that H is isomorphic to the product of several copies of Gm. FromLemma 8.3.4 we obtain:

Corollary 8.3.6. The prestack

GrH,gen ×BunH

pt /H ' Mapsgen(X,H)/H

can be written as a colimit of prestacks, each representable by a projective scheme.

8.3.7. Recall the mapf : QG → QG,G

we claim:

Proposition 8.3.8. The functor f ! : D-mod(QG)→ D-mod(QG,G) is fully faithful and admitsa left adjoint (to be denoted f!).

Proof. By assumption, the group ZG is connected, and so is isomorphic to the product ofseveral copies of Gm. For a test-scheme S and an S-point of QG,G, the groupoid of its liftsto an S-point of QG is a torsor over the group of S-points of Mapsgen(X,ZG). Moreover, thisgroupoid is non-empty in the Zariski topology on S.

Thus, the map f Zariski-locally splits as a product of the base times Mapsgen(X,ZG). Thisimplies that f ! is fully faithful in view of Theorem 8.3.2(a).

To prove the existence of f!, we note that the map f factors as

QG → QG × pt /ZGf ′→ QG,G,

so it suffices to show that the functor

f ′! : D-mod(QG × pt /ZG)→ D-mod(QG,G)

is defined.

Now, the map f ′ Zariski-locally splits as a product of the base times the prestackMapsgen(X,ZG)/ZG, and the assertion follows from Corollary 8.3.6.

Combining Proposition 8.3.8 with diagram (7.5), we obtain:

Corollary 8.3.9. The functor f! sends Whit(G) to Whit(G,G) and provides a left adjoint tof ! : Whit(G,G)→Whit(G).


Remark 8.3.10. It is not difficult to show that the functor f! intertwines the functors AvN,χG∗ ,

acting on D-mod(QG) and D-mod(QG,G), respectively.

8.4. Hecke action.

8.4.1. Fix an integer n, and consider the version of the Hecke stack with n points

BunG .Xn × BunG

Hn

→h

←h

The (ind)-algebraic stack HBunG classifies the data of:

• an n-tuple of points y of X;• a pair of G-bundles PG and P′G;• an isomorphism α between PG and P′G defined away from x.

8.4.2. Recall also that to an n-tuple λ = λ1, ..., λn of elements of Λ+ we associate an object

Sλ ∈ D-mod(BunG)♥[dim(BunG)].

The Verdier dual of Sλ identifies with S−w0(λ)[−2 dim(BunG)], where w0 is the longest ele-ment of the Weyl group of G.

The key property of Sλ is that it is ULA with respect to the projections←h and

→h .

8.4.3. We let Hλ,→ be the functor D-mod(Xn × BunG)→ D-mod(BunG) defined by

F 7→→h∗(←h !(F)

!⊗ Sλ)[2n].

We let Hλ,← be the functor D-mod(BunG)→ D-mod(Xn × BunG) defined by

F 7→←h∗(→h !(F)

!⊗ Sλ).

The ULA property of Sλ and the fact that both maps←h and

→h are schematic and proper

imply:

Lemma 8.4.4. The functor Hλ,→ is the left adjoint of the functor and H−w0(λ),←.

8.4.5. Let Y be a scheme acted on by G. Consider the corresponding prestack

Y := Mapsgen(X,G\Y ) ×Mapsgen(X,pt /G)

Maps(X,pt /G),

equipped with its natural projection to Maps(X,pt /G) = BunG, see Sect. 6.1.1.

It follows from the definitions that we can create a diagram

BunG,Xn × BunG

Hn

YXn × Y

HnY

→h

←h

→hY

←hY


in which both the left and the right diamonds are Cartesian.

We define the Hecke functors

Hλ,→Y : D-mod(Xn × Y)→ D-mod(Y) and Hλ,←

Y : D-mod(Y)→ D-mod(Xn × Y)

to be

F 7→ (→hY )∗((

←hY )!(F)

!⊗ SλY )[2n] and F 7→

←h∗(→h !(F)

!⊗ SλY ),

respectively, where SλY denotes the pullback of Sλ to HnY.

Our key tool will be the following assertion (whose proof is the same as that of Lemma 8.4.4):

Lemma 8.4.6. The functor Hλ,→Y is the left adjoint of H

−w0(λ),←Y .

8.4.7. Taking Y to be one of the prestacks

BunP -genG ;QG,QG,P or Qext,

we obtain the corresponding pairs of adjoint functors

Hλ,→BunP -gen

G

: D-mod(Xn × BunP -genG ) D-mod(BunP -gen

G ) : Hλ,←BunP -gen

G

;

Hλ,→QG

: D-mod(Xn × QG) D-mod(QG) : Hλ,←QG

;

Hλ,→QG,P

: D-mod(Xn × QG,P ) D-mod(QG,P ) : Hλ,←QG,P

;

and

Hλ,→Qext : D-mod(Xn × Qext) D-mod(Qext) : Hλ,←

Qext .

Furthermore, the definitions of Sect. 7 can be adapted to define the full subcategories

D-mod(Xn × BunP -genG )N ⊂ D-mod(Xn × BunP -gen

G );

D-mod(Xn × QG)N,χG ⊂ D-mod(Xn × QG);

D-mod(Xn × QG,P )N,χP ⊂ D-mod(Xn × QG,P )

and

D-mod(Xn × Qext)Next

⊂ D-mod(Xn × Qext).

It follows from the definitions that the above Hecke functors preserve the correspondingsubcategories, i.e., define (automatically, mutually adjoint) functors

D-mod(Xn × BunP -genG )N D-mod(BunP -gen

G )N = I(G,P );

D-mod(Xn × QG)N,χG D-mod(QG)N,χG = Whit(G);

D-mod(Xn × QG,P )N,χP D-mod(QG,P )N,χP = Whit(G,P );

D-mod(Xn × Qext)Next

D-mod(Qext)Next

= Whitext(G).

Furthermore, the Hecke functors commute in the natural sense with the averaging functors

AvN∗ , AvN,χG

∗ , AvN,χP∗ and AvN,χcoeff

∗ .

From here, we obtain that the Hecke functors commute in the natural sense with the functorslisted in Sect. 8.1.1.


8.4.8. Hence, ifG : D-mod(Y1)N,? → D-mod(Y2)N,??

is one of the functors from Sect. 8.1.1, and GL is its (potentially partially defined) left adjoint,which is defined on an object F2 ∈ D-mod(Y2)N,?? is an object, then for any

n, λ and F ∈ D-mod(Xn),

the functor GL is defined also on

Hλ,→Y2

(F F2) ∈ D-mod(Y2)N,??,

and we haveGL Hλ,→

Y2(F F2) ' Hλ,→

Y2(F GL(F2)).

8.5. The functor of Poincare series. In this subsection we will show that the functors

W-coeffG : D-mod(BunG)→Whit(G) and W-coeffG,G : D-mod(BunG)→Whit(G,G)

both admit left adjoints; to be denoted

PoincG and PoincG,G,

respectively.

8.5.1. Recall that

W-coeffG = AvN,χG∗ (rG)! (penh

B )! and W-coeffG,G = AvN,χG∗ (rG,G)! (penh

B )!

Recall (see Sect. 8.2.5) that we have already shown that the functor (penhB )! admits a left

adjoint. We will now show that the functors

AvN,χG∗ (rG)! : D-mod(BunB -gen

G )→Whit(G)

andAvN,χG∗ (rG,G)! : D-mod(BunB -gen

G )→Whit(G,G)

admit left adjoints.

I.e., we have to show that the partially defined functors

(rG)! : D-mod(QG)→ D-mod(BunB -genG ) and (rG,G)! : D-mod(QG,G)→ D-mod(BunB -gen

G )

are actually defined on the full subactegories

Whit(G) ⊂ D-mod(QG) and Whit(G,G) ⊂ D-mod(QG,G),

respectively.

Note that since we know that the functor f!, left adjoint to f ! is well-defined, and f ! isconservative, the assertions for (rG)! and (rG,G)! are equivalent. We will prove one involving(rG)!.

8.5.2. The plan of the proof is the following: we will show that there exists a particular object

ψQ ∈Whit(G),

on which the functor (rG)! is defined for reasons of holonomicity, and that the category Whit(G)is generated by the images of objects of the form

F ψQ, F ∈ D-mod(Xn)

under the Hecke functors

Hλ,→QG

: D-mod(Xn × QG)N,χG →Whit(G).

This would imply that the functor PoincG is defined on all of Whit(G) is view of Sect. 8.4.8.


8.5.3. Recall the (algebraic) stack BunN,ρ(ωX), see Sect. 1.1.3 for the notation. Note that wehave a canonically defined schematic and proper map

mG : BunN,ρ(ωX) → QG.

Recall the object

ψ ∈ D-mod(BunN,ρ(ωX)).

Set

ψQ := (mG)!(ψ).

It follows from the definitions that

ψQ ∈ D-mod(QG)N,χG .

8.5.4. We claim the functor (rG)! is defined on ψQ. Indeed, its suffices to show that the functor

(rG mG)! : D-mod(BunN,ρ(ωX))→ D-mod(BunB -genG ),

left adjoint to the pullback functor (rG mG)!, is defined on ψ.

We note that the morphism rG mG factors as

BunN,ρ(ωX) → BunBkB→ D-mod(BunB -gen

G ),

where kB is schematic and proper. Hence, it remains to show that the !-pushforward along

BunN,ρ(ωX) ' BunB ×BunT

pt→ BunB ,

is defined on ψ.

However, the above morphism is a schematic morphism between algebraic stacks, whileψ ∈ D-mod(BunN,ρ(ωX)) is holonomic. This implies the well-definedness of the !-pushforward.

8.5.5. Thus, for the existence of PoincG, it remains to show that the category Whit(G) isgenerated by the essential images of objects

F ψQ, F ∈ D-mod(Xn)

under the Hecke functors

Hλ,→QG

: D-mod(Xn × QG)N,χG →Whit(G).

This results from Lemma 7.4.6 the following assertion:

Lemma 8.5.6.

(a) The image in QG under the map→hQG of the support of

(Xn × Im(mG)) ×BunG

supp(Sλ) ⊂ HnQG

intersects a dominant stratum

BunB ×BunT

Xµ ⊂ QG

(see (6.1)) only if m ≤ n and Σµi ≤ Σλj.

(b) For m = n and µ = λ, the functor

F ∈ D-mod(Xn) 7→ Hλ,→QG

(F ψQ)|BunB ×

BunT

Xµ


identifies, in terms of the equivalence of Lemma 7.4.6(c) with the functor

D-mod(Xn)→ D-mod(Xn)→ D-mod(

Xµ),

given by restriction to the complement of the diagonal divisor, followed by that of direct image

along the partial symmetriczation mapXn →

Xµ.

8.6. Functors associated to the extended Whittaker category: the case of P = G.The goal of this subsection and the next is to prove the existence of the left adjoint of thefunctor

(iP )! : Whitext(G)→Whit(G,P ).

Namely, we will show that the partially defined left adjoint (iP )! to

(iP )! : D-mod(Qext)→ D-mod(QG,P )

is defined on the full subcategory

Whit(G,P ) ⊂ D-mod(QG,P ).

8.6.1. We shall first consider the case when P = G, and then reduce the general case to thisone. The proof of the fact that the functor (iG)! is defined on

Whit(G,G) ⊂ D-mod(QG,G)

is based on the Hecke action. Namely, as in Sect. 8.5, it suffices that the functor (iG)! is definedon the object

f!(ψQ) ∈ D-mod(QG,G).

I.e., we need to show that the functor (iG f mG)! is defined on ψ ∈ D-mod(BunN,ρ(ωX)).

8.6.2. Note that the morphism

iG f mG : BunN,ρ(ωX) → Qext

can be canonically factored as

BunN,ρ(ωX) → (BunN,ρ(ωX) ×ΠiA1)/T

kext

−→ Qext,

where the map kext is schematic and proper. In the above formula, T acts on ΠiA1 via

T → T/ZG = ΠiGm.

8.6.3. Hence, in order to show that (iG f mG)! is defined on ψ ∈ D-mod(BunN,ρ(ωX)), itsuffices to show that the direct image with compact supports along

BunN,ρ(ωX) → (BunN,ρ(ωX) ×ΠiA1)/T

is defined on ψ.

However, the map in question is between algebraic stacks, and the object ψ is holonomic.This implies the well-definedness of the !-pushforward.


8.7. Functors associated to the extended Whittaker category: the general case. Wewill now prove that the functor

(iP )! : D-mod(QG,P )→ D-mod(Qext)

is defined on the full subcategory

Whit(G,P ) ⊂ D-mod(QG,P )

for any parabolic P . We shall take as an input the fact that this is the case for the functor iM ,i.e., when G is replaced by M .

8.7.1. First, since the functor

(′qP )∗ (′ιP )! : Whit(G,P )→Whit(M,M)

is conservative and admits a left adjoint, it suffices to show that the partially defined composedfunctor

(iP ′ιP )! (′qP )∗ : D-mod(QM,M )→ D-mod(Qext)

is defined onWhit(M,M) ⊂ D-mod(QM,M ).

We will show more generally, that for any F ∈ D-mod(QM,M ), for which the functor (iM )! isdefined on F, the functor (iP ′ιP )! (′qP )∗ is also defined on F.

8.7.2. Let Qext,≤PG ⊂ Qext

G be the closed subfunctor corresponding to the condition that the γi’svanish for i not in the Dynkin diagram of M . The map iP factors as

QG,P → Qext,≤PG → Qext

G ,

where the first arrow is an open embedding.

Let jP denote the composed map

QP,M′ιP−→ QG,P → Q

ext,≤PG .

It suffices to show that the (partially defined) functor (jP )! (′qP )∗ is defined on F as above.

8.7.3. Note that we have a Cartesian diagram

QP,M′iM−−−−→ BunP ×

BunP -genG

Qext,≤PG

′jP−−−−→ Qext,≤PG

′qP

y y′qQPQM,M

iM−−−−→ QextM ,

where the composed horizontal map equals jP .

Now, as in Sect. 8.2, one shows that the functor

(′jP )! (′qQP )∗ : D-mod(QextM )→ D-mod(Qext,≤P

G ),

left adjoint to (′qQP )∗ (′jP )!, is defined.

Finally, we note that the partially defined functors

(′iM )! (′qP )∗ and (′qQP )∗ (iM )!

are canonically isomorphic, because their respective right adjoints

(′qP )∗ (′iM )! and (iM )! (′qQP )∗

are isomorphic by base change.


Part IV: Gluing functors for the extended Whittaker category

9. Gluing functors: the statements

9.1. The first functor. Let P1 ⊂ P2 be a pair of parabolics. Our interest in this section isthe gluing functor

(9.1) (iP1)! (iP2

)! : Whit(G,P2)→Whit(G,P1).

(Unfortunately), we will not be able to “describe” the above functor explicitly. This isprimarily due to the fact that we do not really know if what terms we could give such adescription.

Instead, however, we will able to describe rather explicitly two other functors, obtained from(9.1) by composition.

One of these functors is

PoincP1,P1 (iP1)! (iP2)! : Whit(G,P2)→ I(G,P1).

The other is

(′qP1)∗ (′ιP1

)! (iP1)! (iP2

)! : Whit(G,P2)→Whit(M1,M1).

9.1.1. The description of both these functors uses the following ingredient. Consider the opensubstack

Qext,≥P1

G ⊂ QextG ,

given by the condition that the elements γi do not vanish for i being in the Dynkin diagram ofM1.


QG,P2

iP2−−−−→ Qext,≥P1

G

iP1←−−−− QG,P1ytP1

QG,P1,

where tP1 iP1

is the identity map.

Hence, we obtain a natural transformation

(9.2) D-mod(Qext,≥P1

G )→ D-mod(QG,P1), (iP1

)! → (tP1)!,

where the right-hand is only partially defined. Composing, we obtain a natural transformation

(9.3) (iP1)! (iP2)! → (tP1 iP2)!

If we apply the functors in (9.3) to an object F ∈Whit(G,P2), we have

(iP1)! (iP2

)!(F) ∈ D-mod(QG,P1)N,χP1 = Whit(G,P1).

Hence, by adjunction, we obtain a map

(9.4) (iP1)! (iP2

)!(F)→ AvN(P1)∗ (tP1

iP2)!(F)

In Sects. 10.2 and 10.3 we will prove:


Proposition 9.1.2.

(a) The partially defined functor (tP1 iP2)! : D-mod(QG,P2)→ D-mod(QG,P1) is defined on thefull subcategory Whit(G,P2) ⊂ D-mod(QG,P2).

(b) The natural transformation of functors Whit(G,P2) → Whit(G,P1) given by (9.4) is anisomorphism.

9.1.3. We will now describe the functor

(9.5) Whit(G,P2)(iP1

)!(iP2)!−→ Whit(G,P1)

PoincP1,P1−→ I(G,P1).

Consider the composition

Whit(G,P2)PoincP2,P2−→ I(G,P2)

CTenhP1,P2−→ I(G,P1).

We claim that there is a canonically defined natural transformation between the above func-tors, namely,

(9.6) PoincP1,P1(iP1

)! (iP2)! → CTenh

P1,P2PoincP2,P2

.

Indeed, we first apply the natural transformation (9.4) and obtain a map

(9.7) PoincP1,P1(iP1

)! (iP2)! → PoincP1,P1

AvN(P1)∗ (tP1

iP2)!.

Recall now that PoincP1,P1= (rG,P1

)!, where we remind that rG,P1denotes the map QG,P1

→BunP1 -gen

G .

We recall that

CTenhP1,P2

= AvN(P1)∗ (penh

P1,P2)!,

so

CTenhP1,P2

PoincP2,P2 = AvN(P1)∗ (penh

P1,P2)! (rG,P2)!.

Let us note, however, that the map rG,P2 : QG,P2 → BunP2 -genG factors as

QG,P2

tP1iP2−→ QG,P1

rG,P1−→ BunP1 -genG

penhP1,P2−→ BunP2 -gen

G .

Hence, we rewrite

(9.8) CTenhP1,P2

PoincP2,P2' AvN(P1)

∗ (penhP1,P2

)! (penhP1,P2

)! (rG,P1)! (tP1

iP2)!.

Thus, in order to construct the natural transformation (9.6), we have to construct a naturaltransformation

(9.9) (rG,P1)! AvN(P1)

∗ (tP1 iP2

)! → AvN(P1)∗ (penh

P1,P2)! (penh

P1,P2)! (rG,P1

)! (tP1 iP2

)!.

The latter is obtained as the composition

(rG,P1)! AvN(P1)∗ (tP1 iP2)! → AvN(P1)

∗ (rG,P1)! (tP1 iP2)! →

→ AvN(P1)∗ (penh

P1,P2)! (penh

P1,P2)! (rG,P1)! (tP1 iP2)!,

where the first arrow comes by adjunction from the fact that (rG,P1)! maps

D-mod(QG,P1)N(P1) → D-mod(BunP1 -gen

G )N(P1) = I(G,P1),

and the second arrow comes from the ((penhP1,P2

)!, (penhP1,P2

)!) adjunction.

We will prove:

Theorem 9.1.4. The natural transformation (9.6) is an isomorphism.


It is the description of the functor PoincP1,P1(iP1

)! (iP2)! in terms as the functor

CTenhP1,P2

PoincP2,P2 that we mean to be an explicit description of the former functor.

9.2. The second functor. The goal of this subsection we will state a theorem that describesexplicitly the functor

(9.10) (′qP1)∗ (′ιP1

)! (iP1)! (iP2

)! : Whit(G,P2)→Whit(M1,M1).

9.2.1. First, we will reduce the study of this functor to the case when P2 = G. This will usethe following ingredient:

Note that we have a Cartesian diagram

QG,P2

iP2−−−−→ Qext,≤P2

G

′ıP2

x x′jP2

QP2,M2

′iM2−−−−→ BunP2 ×Bun

P2 -gen

G

Qext,≤P2

G

′qP2

y y′qQP2

QM2,M2

iM2−−−−→ QextM2,

from which we obtain a natural transformation

(9.11) (′qQP2)∗ (′iM2

)! (′ıP2)! → (′qQP2

)∗ (′jP2)! (iP2

)!, Whit(G,P2)→Whitext(M2).

In Sect. 10.1 we will prove:

Proposition 9.2.2. The natural transformation (9.11) is an isomorphism.

9.2.3. Note that the composition

QP1,M1

ιP1−→ QG,P1

iP1−→ QextG

canonically factors as

QP1,M1→ BunP2

×Bun

P2 -gen

G

QG,P1

id×iP1−→ BunP2×

BunP2 -gen

G

Qext,≤P2

G

′jP2−→ Qext,≤P2

G → QextG ,

and where we have a Cartesian diagram

QP1,M1

′′ıP1(M2)−−−−−−→ BunP2×

BunP2 -gen

G

QG,P1

id×iP1−−−−−→ BunP2×

BunP2 -gen

G

Qext,≤P2

Gy y y′qQP2

QP1(M2),M1

′ıP1(M2)−−−−−→ QM2,P1(M2)

iP1(M2)−−−−−→ QextM2

′qP1(M2)

yQM1,M1 .

Hence, we can rewrite the functor (′qP1)∗ (′ιP1

)! (iP1)! (iP2

)! of (9.10) as

(′qP1)∗ (′′ıP1(M2))

! (id×iP1)! (′jP2

)! (iP2)! '

' (′qP1(M2))∗ (′ıP1(M2))! (iP1(M2))

! (′qQP2)∗ (′jP2)! (iP2)!.


Using Proposition 9.2.2, we further rewrite it as

(′qP1(M2))∗ (′ıP1(M2))! (iP1(M2))

! (′qQP2)∗ (′iM2

)! (′ıP2)!,

and finally, using the fact that for Whittaker sheaves

(′qQP2)∗ (′iM2)! ' (iM2)! (′qP2)∗,

we obtain

(9.12) (′qP1)∗(′ιP1)!(iP1)!(iP2)! ' (′qP1(M2))∗(′ıP1(M2))!(iP1(M2))

!(iM2)!(′qP2

)∗(′ıP2)!.

I.e., we obtain that the functor of (9.10) identifies with the composition of

(′qP2)∗ (′ıP2)!, Whit(G,P2)→Whit(M2,M2),

and the functor

(′qP1(M2))∗ (′ıP1(M2))! (iP1(M2))

! (iM2)! : Whit(M2,M2)→Whit(M1,M1),

which is nothing but the functor (9.10) for the reductive group M2 and its parabolic subgroupP1(M2).

9.2.4. Thus, our current task is to describe the functor

(9.13) (′qP )∗ (′ιP )! (iP )! (iG)! : Whit(G,G)→Whit(M,M).

Consider the fiber product

QG,P ×BunG

BunP− ,

where P− is the parabolic opposite to P . Let

ZastgenP ⊂ QG,P ×

BunGBunP−

denote the open substack, given by the condition that the generic reduction to B (which is partif the data for a point of QG,P ) and the reduction to P− are transversal at the generic point ofthe curve.

Let uP denote the projection ZastgenP → QG,P . Note now that the tautological map

ZastgenP → BunP− → BunM

naturally lifts to a map

ZastgenP → QM,M .

We denote this map by vP .


ZastgenP ×

QG,P

QG,G,

where QG,G → QG,P is the map tP iG. Denote the maps

QM,M ← ZastgenP ×

QG,P

QG,G → QG,G

by ′uP and ′vP , respectively.

Consider the functor

(′uP )∗ (′vP )! : D-mod(QM,M )→ D-mod(QG,G),

and its partially defined left adjoint (′vP )! (′uP )∗.


We claim that there is a canonically defined natural transformation of partially definedfunctors

(9.14) (′qP )∗ (′ιP )! (iP )! (iG)! → (′vP )! (′uP )∗, D-mod(QG,G)→ D-mod(QM,M ).

9.2.6. To define the map (9.14), we first apply the natural transformation (9.4) (which, accord-ing to Proposition 9.1.2, is in fact an isomorphism when evaluated on objects of Whit(G,G)):

(9.15) (′qP )∗ (′ιP )! (iP )! (iG)! → (′qP )∗ (′ιP )! AvN(P)∗ (tP iG)!.

Note that

(′qP )∗(′ιP )!AvN(P)∗ (tP iG)! ' (′qP )∗AvN(P)

∗ (′ιP )!(tP iG)! ' (′qP )∗(′ιP )!(tP iG)!.

We now claim that there is a canonically defined natural transformation

(9.16) (′qP )∗ (′ιP )! → (vP )! (uP )∗, D-mod(QG,P )→ D-mod(QM,M ).

Once the natural transformation (9.16) is defined, the sought-for natural transformation(9.14) is set to be

(′qP )∗ (′ιP )! (iP )! (iG)! → (′qP )∗ (′ιP )! (tP iG)! →→ (uP )! (vP )∗ (tP iG)! ' (′vP )! (′uP )∗.

9.2.7. The natural transfrmation (9.16) is defined as follows. To specify it, by adjunction, wecan equivalently specify a natural transformation

(9.17) (′qP )∗ (′ιP )! (uP )∗ (vP )! → Id .

The natural transformation (′qP )∗ (′ιP )! (uP )∗ (vP )! is given by pull-push along theCartesian diagram

(9.18)

ZastgenP ×

QG,P

QP,M −−−−→ QP,M′qP−−−−→ QM,My y′ιP

ZastgenP

uP−−−−→ QG,P

vP

yQM,M .

Note however, that the open embedding BunM → BunP ×BunM

BunP− give rise to an open

embedding

QM,M → ZastgenP ×

QG,P

QP,M .

This gives rise to the natural transformation (9.17). This natural transformation is the keystep in the construction of the natural transformation (9.14).


Proposition 9.2.8. Both sides of (9.16) are well-defined, and the natural transformation(9.16) is an isomorphism.


9.2.9. Thus, from Proposition 9.2.8 we obtain:

Theorem 9.2.10. The natural transformation

(′qP )∗ (′ιP )! (iP )! (iG)! → (′vP )! (′uP )∗

is an isomorphism when evaluated on objects of Whit(G,G) ⊂ D-mod(BunG).

We regard the right-hand side in Theorem 9.2.10 as giving a more explicit description of thefunctor (′qP )∗ (′ιP )! (iP )! (iG)!, thus accomplishing the goal of this subsection.

9.3. Relation between Theorems 9.1.4 and 9.2.10.

9.3.1. Let us summarize the previous discussion. For a pair of parabolics P1 ⊂ P2 we considerthe functor

i!P1 (iP2

)! : Whit(G,P2)→Whit(G,P1),

and the compositions

PoincP1,P1i!P1 (iP2

)! : Whit(G,P2)→ I(G,P1)

and(′qP1

)∗ (′ιP1)! i!P1

(iP2)! : Whit(G,P2)→Whit(M1,M1).

In this subsection we will study the further compositions

(9.19) (ıM1)! PoincP1,P1

i!P1 (iP2

)! : Whit(G,P2)→ D-mod(BunM1)

and

(9.20) PoincM1,M1(′qP1

)∗ (′ιP1)! i!P1

(iP2)! : Whit(G,P2)→ D-mod(BunM1

).

9.3.2. Recall that in Sect. 9.1 we have constructed a natural transformation

PoincP1,P1i!P1 (iP2

)! → CTenhP1,P2

PoincP2,P2,

and stated Theorem 9.1.4 that this natural transformation is an isomorphism.

Hence, composing with the functor (ıM1)! : I(G,P1)→ D-mod(BunM1

), we obtain a naturaltransformation (in fact, an isomorphism, according to Theorem 9.1.4):

(9.21) (ıM1)! PoincP1,P1

i!P1 (iP2

)! → (ıM1)! CTenh

P1,P2PoincP2,P2

'

' CTP1(M2) (ıM2)! PoincP2,P2

,

where CTP1(M2) is the Constant term functor D-mod(BunM2)→ D-mod(BunM1).

9.3.3. Recall also that in Sect. 9.2 we have constructed a natural transformation

(′qP1)∗ (′ιP1)! (iP1)! (iP2)! → (′vP1(M2))! (′uP1(M2))∗ (′qP2

)∗ (′ıP2)!,

and Theorem 9.2.10 asserted that this natural transformation is an isomorphism.

Hence, composing with the functor PoincM1,M1: Whit(M1,M1) → D-mod(BunM1

), weobtain a natural transformation (in fact, an isomorphism according to Theorem 9.2.10):

(9.22) PoincM1,M1 (′qP1)∗ (′ιP1)! i!P1 (iP2)! →

' PoincM1,M1 (′vP1(M2))! (′uP1(M2))∗ (′qP2

)∗ (′ıP2)! '

' (penhB(M1) rM1,M1

′vP1(M2))! (′uP1(M2))∗ (′qP2

)∗ (′ıP2)!,

where we note that the map penhB(M1) rM1,M1

′vP1(M2) is just the projection

ZastgenP1(M2) ×

QM2,P1(M2)

QM2,M2→ Zastgen

P1(M2) → QM2,M2→ BunM2

.


9.3.4. We will now complete the natural transformations (9.21) and (9.22) to a commutativediagram of functors and natural transformations:

(9.23)

(ıM1)! PoincP1,P1

i!P1 (iP2

)!

CTP1(M2) (ıM2)! PoincP2,P2

PoincM1,M1 (′qP1)∗ (′ιP1)! i!P1 (iP2)!

(penhB(M1) rM1,M1

′vP1(M2))! (′uP1(M2))∗ (′qP2

)∗ (′ıP2)!

--

--

OO

OO

In order to construct the vertical arrows in this diagram we need the following observation.

9.3.5. Note that for a parabolic P we have a natural transformation

(9.24) PoincM,M (′qP )∗ (′ıP )! → (ıM )! PoincP,P , Whit(G,P )→ D-mod(BunM ).

Namely, it comes from the Cartesian diagram

QM,M

′qP←−−−− QP,M′ıP−−−−→ QG,P

rM,M

y rP,M

y yrG,P

BunMqP←−−−− BunP

ıP−−−−→ BunP−genG .


Proposition 9.3.6. The natural transformation (9.24) is an isomorphism.

9.3.7. We let the left vertical arrow in the diagram (9.23) to be given by the map (9.24) forP = P1, precomposed with i!P1

(iP2)!.

The right vertical arrow in the diagram (9.23) is the composition of

CTP1(M2) PoincM2,M2(′qP2

)∗ (′ıP2)! → (ıM2

)! → CTP1(M2) (ıM2)! PoincP2,P2

,

given by the map (9.24) for P = P2 composed with CTP1(M2), and a map

(penhB(M1) rM1,M1 ′vP1(M2))! (′uP1(M2))

∗ (′qP2)∗ (′ıP2

)! →

→ CTP1(M2) PoincM2,M2 (′qP2)∗ (′ıP2)!,

obtained by composing (′qP2)∗ (′ıP2)! with a natural transformation

(9.25) (penhB(M1) rM1,M1 ′vP1(M2))! (′uP1(M2))

∗ → CTP1(M2) PoincM2,M2 ,

described below.

9.3.8. Note that in constructing the natural transformation (9.25), we are working with thereductive group M2 and its parabolic P1(M2). So, it is enough to construct this natural trans-formation for G and a parabolic P . I.e., we want to construct a natural transformation

(9.26) (penhB rM,M ′vP )! (′uP )∗ → CTP PoincG,G .

The crucial ingredient is provided by the natural transformation (in fact, an isomorphism)of [DrGa4]:

CTP ' (q−P )! (p−P )∗.


Thus, the right-hand side in (9.26) is given by *-pull and !-push along the diagram

BunP− ×BunG

QG,G −−−−→ QG,Gy yBunP−

pP−−−−−→ BunG

qP−

yBunM .

Now, the left-hand side in (9.26) is given by *-pull and !-push along the diagram

ZastgenP ×

QG,P

QG,G −−−−→ QG,GyZastgen

PyBunP−

qP−

yBunM .

The require natural transformation comes now from the fact that we have an open embedding

ZastgenP ×

QG,P

QG,G → BunP− ×BunG

QG,G,

which comes by base change from the open embedding

ZastgenP → QG,P ×

BunGBunP− .

The next assertion is proved in a manner parallel to that of the isomorphism in Sect. 3.1.4:

Lemma 9.3.9. The natural transformation (9.26) is an isomorphism.

9.3.10. Thus, we have constructed all the arrows in the diagram (9.23). As in (3.4) we have:

Lemma 9.3.11. The diagram (9.23) is commutative.

9.3.12. Let us for a moment assume Proposition 9.3.6. We obtain:

Corollary 9.3.13. Theorem 9.1.4 implies Theorem 9.2.10.

Proof. It is enough to show that the natural transformation

PoincP1,P1 i!P1 (iP2)! → CTenh

P1,P2PoincP2,P2

becomes an isomorphism after applying the functor (ıM1)!, since the latter is conservative. I.e.,

it is enough to show that the natural transformation (9.21) is an isomorphism.

This natural transformation is the top slanted arrow in the diagram (9.23). Now, Theo-rem 9.2.10 implies that the bottom slanted arrow in (9.23) is an isomorphism.

Now, Proposition 9.3.6 implies that the left vertical arrow in (9.23) is an isomorphism.Finally, Proposition 9.3.6 and Lemma 9.3.9 show that the right vertical arrow in (9.23) is alsoan isomorphism.


10. Gluing functors: proofs

In this section we will prove the following assertions stated in Sect. 9: Propositions 9.1.2,9.2.2, 9.3.6 and 9.2.8.

Proposition 9.1.2 will be a rather general assertion about the interplay between Whittakerand Kirillov patterns.

We will group these assertions according to the ideas involved in their proofs.

Namely, Propositions 9.2.2 and 9.3.6 will be proved using a ULA consideration akin to oneused in Sect. 8.2.

Proposition 9.2.8 will be an application of Braden’s theorem, similar to the proof of Theo-rem 2.2.3 from Part I of the paper.

10.1. Proof of Propositions 9.2.2 and 9.3.6: a ULA game.

10.1.1. Consider the algebraic stack

BunP ×BunP -gen

G

BunP ,

and let pr1 and pr2 denote its projections to BunP and BunP , respectively.

Consider the object

(pr1)! (ιP )!(kBunP ) ∈ D-mod(BunP ×BunP -gen

G

BunP ).

The key tool in this subsection is the following assertion, which is proved in the same wayas [BG1, Theorem ???]:

Proposition 10.1.2. The object (pr1)! (ιP )!(kBunP ) is ULA with respect to the map

qP pr1 : BunP ×BunP -gen

G

BunP → BunM .

We will now prove Proposition 9.3.6, and leave Proposition 9.2.2 as its proof is similar.

10.1.3. Since the essential image of the functor

(′ıP )! (′qP )∗ : Whit(M,M)→Whit(G,P )

generates the target, it is enough to show that the natural transformation

PoincM,M (′qP )∗ (′ıP )! (′ıP )! (′qP )∗ → (ıM )! PoincP,P (′ıP )! (′qP )∗

is an isomorphism.

This is equivalent to showing that the natural transformation

(10.1) (rP,M )! (′ıP )! (′ıP )! (′qP )∗ → (ıP )! (ıP )! (qP )∗ (rM,M )!

that comes from the Cartesian diagram

QM,M

′qP←−−−− QP,M′ıP−−−−→ QG,P

′ıP←−−−− QP,M

rM,M

y rP,M

y yrG,P

yrM,M


ıP−−−−→ BunP -genG

ıP←−−−− BunP

is an isomorphism. We will show that (10.1) is an isomorphism for any object F ∈ D-mod(QM,M )on which the functor (rM,M )! is defined.


10.1.4. Recall the map kP : BunP → BunP -genG . We have:

(ıP )! (ıP )! ' (ıP )! (kP )! (ιP )!,

and further, using the properness of the maps kP and

pr2 : BunP ×BunP -gen

G

BunP → BunP

as

(pr2)! (pr1)! (ιP )!.

Hence, we can rewrite the functor (ıP )! (ıP )! (qP )∗ as the composition

(pr2)! (pr1)! (ιP )! q∗,

taken along the diagram

(10.2)


ıP−−−−→ BunPxpr1

BunP ×BunP -gen

G

BunPypr2

BunP .

10.1.5. Recall now the map

kQP : QM,M ×

BunMBunP → QG,P .

Consider now the fiber product

QM,M ×BunM

(BunP ×BunP -gen

G

BunP ),

formed using the map

BunP ×BunP -gen

G

BunP → BunP → BunM .

The key observation is that the projection

QM,M ×BunM

(BunP ×BunP -gen

G

BunP )→ BunP ×BunP -gen

G

BunP → BunP

canonically lifts to a map

QM,M ×BunM

(BunP ×BunP -gen

G

BunP )→ QM,M ×BunM

BunP ' QP,M


QM,M ×BunM

(BunP ×BunP -gen

G

BunP )′kQP−−−−→ QP,M

id× pr1

y y′ıPQM,M ×

BunMBunP

kQP−−−−→ QG,P

Cartesian.


Hence, we rewrite

(′ıP )! (′ıP )! ' (′kQP )! (id×pr1)! (id×ıP )!,

where id×ıP is the map

QP,M ' QM,M ×BunM

BunP → QM,M ×BunM

BunP .

Hence, we can rewrite (′ıP )! (′ıP )! (′qP )∗ as the composition

(′kQP )! (id×pr1)! (id×ıP )! (id×qP )∗

taken along the diagram

(10.3)

QM,Mid×qP←−−−− QM,M ×

BunMBunP

id×ıP−−−−→ QM,M ×BunM

BunPxid× pr1

QM,M ×BunM

(BunP ×BunP -gen

G

BunP )y′kQP

QM,M ×BunM

BunP .

10.1.6. Comparing the diagrams (10.2) and (10.3), and using the commutative diagram

QM,M ×BunM

(BunP ×BunP -gen

G

BunP )′kQP−−−−→ QM,M ×

BunMBunP

rM,M×id

y yrM,M×id

BunP ×BunP -gen

G

BunPpr2−−−−→ BunP ,

we obtain that in ordet to show that (10.1) is an isomorphism, it is enought to show that thenatural transfrmation

(10.4) (rM,M × id)! (id×pr1)! → (pr1)! (rM,M × id)!,

coming from the Cartesian diagram

QM,M ×BunM

BunPid× pr1←−−−−− QM,M ×

BunM(BunP ×

BunP -genG

BunP )

rM,M×id

y yrM,M×id

BunPpr1←−−−− BunP ×

BunP -genG

BunP ,

is an isomorphism, when evaluated on objects of the form

(id×ıP )! (id×qP )∗(F), F ∈ D-mod(QM,M ).

10.1.7. Note that the ULA property of ı!(kBunP ) ∈ D-mod(BunP ) with respect to the morphismq implies that

(id×ıP )! (id×qP )∗(F) ' (id×qP )!(F)!⊗ (rM,M × id)!(ı!(kBunP ))[2 dim(BunM )].


10.1.8. Consider the following general situation. Let Y be a smooth algebraic stack, and let

f : W2 →W1

be a map of algebraic stacks over Y. Let G ∈ D-mod(W1) have the property that it is ULA overY, and also f !(G) ∈ D-mod(W2) is ULA over Y.

Let g : Z′ → Z′′ be an arbitrary map of prestacks over Y. Let F ∈ D-mod(Z′) be such thatg!(F) is defined.

Consider the Cartesian diagram

Z′ ×YW1

id×f←−−−− Z′ ×YW2

g×id

y yg×id

Z′′ ×YW1

id×f←−−−− Z′′ ×YW2,

and the resulting map

(10.5) (g × id)! (id×f)!(F!⊗ G)→ (id×f)! (g × id)!(F

!⊗ G).

We have:

Lemma 10.1.9. Under the above circumstances, the map (10.5) is an isomorphism.

Lemma 10.1.9 implies that (10.4) using Proposition 10.1.2.

Proof of Lemma 10.1.9. We have a commutative diagram

(g × id)! (id×f)!(F!⊗ G) −−−−→ (id×f)! (g × id)!(F

!⊗ G)

∼y y

(g × id)!(F!⊗ f !(G)) (id×f)!(g!(F)

!⊗ G)y ∼

yg!(F)

!⊗ f !(G)

id−−−−→ g!(F)!⊗ f !(G).

Hence, it is enough to show that g : Z′ → Z′ and F ∈ D-mod(Z′) as in the lemma, for analgebraic stack W over Y and G ∈ D-mod(W), which is ULA over Y, the map

(g × id)!(F!⊗ G)→ g!(F)

!⊗ G

is an isomorphism. This in turn follows from the commutative diagram

(g × id)!(F!⊗ G) −−−−→ g!(F)

!⊗ G

∼y ∼

y(g × id)!(F

∗⊗ G)[−2 dim(Y)]

∼−−−−→ g!(F)∗⊗ G[−2 dim(Y)].

10.2. Proof of Proposition 9.1.2: a reduction step.


10.2.1. First, as in Sects. 9.2.1 and 9.2.3, one reduces the assertion to the case when P2 = G.Henceforth, we shall denote P1 simply by P .

Next, using the Hecke action as in Sect. 8.5, we obtain that it is enough to establish bothassertions of the proposition when evaluated on a single object of Whit(G,G), described below.

10.2.2. Denote QG,G := QG,G. Recall the map

mG : BunN,ρ(ωX) → QG,

and let denote by the same symbol mG the map

QG,G → QG,G.

The map mG is schematic and proper. The required object of QG,G is

(mG)!(ψ),

where by a slight abuse of notation we denote by ψ the object of D-mod(QG,G) which pulls

back to the same-named object of D-mod(BunN,ρ(ωX)) under the map

BunN,ρ(ωX) → QG,G.

10.2.3. Consider now the algebraic stack QG,P := BunN,ρ(ωX)/ZM and the morphism

mP : QG,P → QG,P .

The morphism mP is also schematic and proper.

Note now that we have a commutative diagram

QG,GmG−−−−→ QG,G

fP

y ytP iG

QG,PmP−−−−→ QG,P ,

where the map fP is schematic and proper.

This implies the statement of Proposition 9.1.2(a): the object ψ ∈ D-mod(QG,G) is holo-nomic, while

(tP iG)! (mG)!(ψ) ' (mP )! (fP )!(ψ).

10.2.4. To prove Proposition 9.1.2(a), we note that the morphism mG can be canonically ex-tended to a morphism

mextG : QG,G × (A1)IG/(Gm)IG → Qext

G ,

which is also schematic and proper.


Furthermore, we have a Cartesian diagram(QG,G × (A1 − 0)IM

)/(Gm)IG

mP−−−−→ QG,Py yiP(QG,G × (A1)IG−IM × (A1 − 0)IM

)/(Gm)IG

mextG−−−−→ Q

ext,≥PGy y(

QG,G × (A1)IG)/(Gm)IG

mextG−−−−→ Qext

G ,x xiG(QG,G × (A1 − 0)IG

)/(Gm)IG

mG−−−−→ QG

where the top left vertical map corresponds to 0 → (A1)IG−IM , and where we identify(QG,G × (A1 − 0)IM

)/(Gm)IG '

(QG,G

)/(Gm)IG−IM ' QG,P

and (QG,G × (A1 − 0)IG

)/(Gm)IG ' QG,G.

10.2.5. Thus, we have a diagram (QG,G × (A1 − 0)IM

)/(Gm)IGyi(

QG,G × (A1 − 0)IG)/(Gm)IG

j−−−−→(QG,G × (A1)IG−IM × (A1 − 0)IM

)/(Gm)IGyt(

QG,G × (A1 − 0)IM)/(Gm)IG ,

which is equipped with a proper map to the diagram

QG,PyiP

QG,GiG−−−−→ Q

ext,≥PGytP

QG,P .

As in (9.4), we have a canonically defined natural transformation

(10.6) i! j! → AvN,χP∗ (t j)!, D-mod(QG,G)N,χG → D-mod(QG,P )N,χP .

To prove Proposition 9.1.2(a), it suffices to show that the map (10.6) is an isomorphism when

evaluated on ψ ∈ D-mod(QG,G)N,χG .

Remark 10.2.6. In fact, it is easy to see that the category D-mod(QG,G)N,χG is canonically

equivalent to Vect with ψ being the generator.

10.3. Proof of Proposition 9.1.2: Kirillov vs. Whittaker. In this subsection we willprove that the map (10.6) is an isomorphism.


10.3.1. Unwinding the definitions, we arrive to the following set-up. Let Y be a prestack,equipped with an action of the group

(Gm nGa)I ,

where I is some finite set. (In our situation we will take I = IG − IM ).

Consider (A1)I as the variety of group-homomorphisms (Ga)I → Aa; in particular, we havea distinguished point χ := 1I ∈ (A1)I .

Consider the diagram

Y ' (Y× (A1 − 0)I)/(Gm)Ij−−−−→ (Y× (A1)I)/(Gm)I

i←−−−− Y/(Gm)IytY/(Gm)I .

It makes sense to consider the full subcategories

D-mod(Y× (A1)I)(Ga)I ,χext

⊂ D-mod(Y× (A1)I)

and (D-mod(Y× (A1)I)(Ga)I ,χext

)(Gm)I

⊂ D-mod((Y× (A1)I)/(Gm)I),

and the corresponding categories on Y× (A1 − 0)I and Y.

Note that

(10.7)(

D-mod(Y× (A1 − 0)I)(Ga)I ,χext)(Gm)I

' D-mod(Y)(Ga)I ,χ,

D-mod(Y)(Ga)I ,χext

' D-mod(Y)(Ga)I ,(D-mod(Y)(Ga)I ,χext

)(Gm)I

' D-mod(Y)(GmnGa)I .

10.3.2. As in (9.4), we have the partially defined functors and a natural transformation

(10.8) i! → Av(Ga)I

∗ t!, D-mod(Y× (A1 − 0)I)(Ga)I ,χext

→ D-mod(Y)(Ga)I .

Proposition 10.3.3. The natural transformation (10.8) evaluated to an isomorphism.

This proposition will imply the isomorphism (10.6).

Remark 10.3.4. Note that the natural transformation (10.8) upgrades to a natural transforma-tion of functors

(10.9)(


→(

D-mod(Y)(Ga)I ,χext)(Gm)I

'

' D-mod(Y)(GmnGa)I .

As such it can be interpreted as the composition(D-mod(Y× (A1 − 0)I)(Ga)I ,χext

)(Gm)I

' D-mod(Y)(Ga)I ,χ → D-mod(Y)Av

(Gm)I

!−→

→ D-mod(Y)(Gm)I Av(Ga)I

∗−→ D-mod(Y)(GmnGa)I .


10.3.5. The proof of Proposition 10.3.3 relies on the following observation. Let act denote theaction map

Y× (Ga)I → Y.

Note now that the functor

(10.10) D-mod(Y)act!

−→ D-mod(Y× (Ga)I)FD→ D-mod(Y× (A1)I)

defines an equivalence

D-mod(Y)→ D-mod(Y× (A1)I)(Ga)I ,χext

,

where FD is the functor of Fourier-Deligne transform along (Ga)I .

Proof of Proposition 10.3.3. We precompose both functors with the equialence (10.10). Thecomposition

D-mod(Y)act!

−→ D-mod(Y×Ga)I)FD→ D-mod(Y× (A1)I)

t!−→ D-mod(Y)

is the identity functor. Hence, its further composition with Av(Ga)I

∗ is the functor Av(Ga)I

∗ .

Now, the composition

D-mod(Y)act!

−→ D-mod(Y× (Ga)I)FD→ D-mod(Y× (A1)I)

i!−→ D-mod(Y)

is manifestly the functor Av(Ga)I

∗ .

Thus, the natural transformation (10.8) is an endomorphism of the functor Av(Ga)I

∗ . Fur-

thermore, this endomorphism acts as the identity on D-mod(Y)(Ga)I ⊂ D-mod(Y). Hence, it iscanonically isomorphic to the identity map.

Remark 10.3.6. Denote Whit(Y) := D-mod(Y)(Ga)I ,χ. Let us denote by Kir∗(Y) (resp., Kir!(Y))

the full subcategory of D-mod(Y)(Gm)I consisting of objects F such that Av(Ga)I

∗ (F) = 0 (resp.,

Av(Ga)I

! (F) = 0).

Note that the equivalence (10.10) gives rise to an equivalence

D-mod(Y)(Gm)I '(

D-mod(Y× (A1)I)(Ga)I ,χext)(Gm)I

.

The above equivalence gives rise to an identification of Kir∗(Y) with the full subcategory of(D-mod(Y× (A1)I)(Ga)I ,χext

)(Gm)I

consisting of objects F′ such that i!(F′) = 0. The functor

j∗ identifies the above category with(


.

Thus, we obtain that (10.7) defines an equivalence

Whit(Y) ' Kir∗(Y).

Explicitly, this equivalence is given by

F ∈Whit(Y) 7→ Av(Gm)I

∗ (F) ∈ Kir∗(Y), F ∈ Kir∗(Y) 7→ Av(Ga)I ,χ∗ (F)[2|I|] ' Av

(Ga)I ,χ! (F).

Simillary, Kir!(Y) identifies with the full subcategory of(

D-mod(Y× (A1)I)(Ga)I ,χext)(Gm)I

consisting of objects F′ such that i∗(F′) = 0. The functor j! identifies the above category with


the full subcategory of(


, consisting of objects F′′, for

which j!(F′′) is defined.

Thus, we have a fully faithful embedding Kir!(Y) →Whit(Y), given by

F ∈ Kir!(Y) 7→ Av(Ga)I ,χ∗ (F) ' Av

(Ga)I ,χ! (F)[−2|I|].

Its essential image consists of those objects F ∈Whit(Y), for which Av(Gm)I

! is defined, and

F 7→ Av(Gm)I

! (F)

provides the inverse functor.

In terms of these identifications, the functor (10.9) can be interpreted as the composition

Kir!(Y) → D-mod(Y)(Gm)I Av(Gm)I

∗−→ D-mod(Y)(GmnGa)I .

I.e., we are applying to an object of Kir!(Y) “the other” averaging functor with respect to (Ga)I ,i.e., one does does not vanish.

10.4. Proof of Proposition 9.2.8. The idea of the proof is to replace the prestacks appearingin the diagram (9.18) by algebraic stacks, and then apply a version of Braden’s theorem a laTheorem 2.2.3.

10.4.1. Let Div+ be the scheme of effective divisors on X (i.e., the disjoint union of symmetricpowers of X). Let Div+

ωX be the following twisted version of Div+: it is the scheme that classifiespairs (L,L→ ωX), where L is a line bundle on X and L→ ωX is a non-zero map.

Consider the forgetful map

Div+×Div+ωX → BunGm , (D,L→ ωX) 7→ L(−D)→ ωX .

Consider the following algebraic stacks:

Q′G,P := BunB ×BunT/ZM

(Div+×Div+

ωX

)IM,

where we identify T/ZM ' (Gm)IM using the simple roots of M .

Q′P,M :=

(BunP ×

BunMBunB(M)

)×

BunT/ZM

(Div+×Div+

ωX

)IM.

Q′M,M := BunB(M) ×BunT/ZM

(Div+×Div+

ωX

)IM.

We let Zast′P be the corresponding open substack of Q′G,P ×BunG

BunP− .


10.4.2. We have a commutative diagram

(10.11)

Zast′P ×Q′G,P

Q′P,M −−−−→ Q′P,M

′q′P−−−−→ Q′M,My y′ι′PZast′P

u′P−−−−→ Q′G,P

v′P

yQ′M,M .

The diagram (10.11) maps to the diagram

ZastgenP ×

QG,P

QP,M −−−−→ QP,M′qP−−−−→ QM,My y′ιP

ZastgenP


vP

yQM,M .

of (9.18) by means of schematic, proper and surjective maps.

Furthermore, the squares

Q′P,M

′ι′P−−−−→ Q′P,Gy yQP,M

′ιP−−−−→ QP,G

and

Zast′Pu′P−−−−→ Q′G,Py y

ZastgenP


are Cartesian.

Hence, in order to prove Proposition 9.2.8, it suffices to prove the following:

Proposition 10.4.3. The naturalt ransformation

(10.12) (′q′P )∗ (′ι′P )! → (v′P )! (u′P )∗, D-mod(Q′G,P )→ D-mod(Q′M,M )

that arises by adjunction from the natural transformation

(′q′P )∗ (′ι′P )! (u′P )∗ (v′P )! → Id

is an isomorphism.


10.4.4. The proof of Proposition 10.4.3 is completely parallel to the proof of Theorem 2.2.3.Namely, we choose a dominant regular co-character γ : Gm → ZM , which defines an action ofGm on the stack BunB over BunT , an in particular over Q′G,P .

Now, as in Sect. 5 one shows that the diagram (10.11) identifies with

Repel(Q′G,P ) ×Q′G,P

Attr(Q′G,P ) −−−−→ Attr(Q′G,P ) −−−−→ Fixed(Q′G,P )y yRepel(Q′G,P ) −−−−→ Q′G,PyFixed(Q′G,P ).

This allows to reduce the assertion of Proposition 10.4.3 to the situation of the usual Bradentheorem.


References

[Bar][BD] A. Beilinson and V. Drinfeld, Hitchin.

[BG1] A. Braverman, D Gaitsgory, Geometric Eisenstein Series.

[BG2] A. Braverman, D Gaitsgory, Deformations of Eisenstein Series.[BFGM] IC of Drinfeld.

[Dr] On Gm-actions.

[DrGa1] On some finiteness questions.[DrGa2] Compact generation.

[DrGa3] Braden.[DrGa4] Constant term functor(s).

[DS]

[FGV] Whittaker patterns.[Ga] Contractibility

[Ras] PhD Thesis

[Sch] S. Schieder.

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