arX

iv:2

112.

0872

5v1

[m

ath.

QA

] 1

6 D

ec 2

021

WHITTAKER MODULES FOR gl AND W1+∞–MODULES WHICH ARE

NOT TENSOR PRODUCTS

DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

Abstract. We consider the Whittaker modules M1(λ,µ) for the Weyl vertex algebra M

(also called βγ vertex algebra), constructed in [4], where it was proved that these modulesare irreducible for each finite cyclic orbifold MZn . In this paper, we consider the modulesM1(λ,µ) as modules for the Z–orbifold of M , denoted by M0. M0 is isomorphic to thevertex algebra W1+∞,c=−1 = M(2)⊗M1(1) which is the tensor product of the Heisenbergvertex algebra M1(1) and the singlet algebra M(2) (cf. [2], [14], [19]). Furthermore, these

modules are also modules of the Lie algebra gl with central charge c = −1. We prove that

they are reducible as gl–modules (and therefore also as M0–modules), and we completelydescribe their irreducible quotients L(d,λ,µ). We show that L(d,λ,µ) in most cases arenot tensor product modules for the vertex algebra M(2) ⊗ M1(1). Moreover, we showthat all constructed modules are typical in the sense that they are irreducible for theHeisenberg-Virasoro vertex subalgebra of W1+∞,c=−1.

Contents

1. Introduction 2Free field realization of L(d,λ,µ). 3Typical W1+∞,c–modules 4A connection with Whittaker gl(2ℓ)–modules. 42. Preliminaries 52.1. Classical Whittaker modules 52.2. Generalized Whittaker modules 52.3. Example: g = gl(2ℓ,C) 6

2.4. Whittaker modules for g = gl 63. Weyl vertex algebra and its Whittaker modules 74. W1+∞-algebra at central charge c = −1 and its Whittaker modules 104.1. The W1+∞-algebra approach 10

4.2. Approach using the Lie algebra gl 10

4.3. Connection between gl–modules and W1+∞–modules 11

5. M1(λ,µ) as a Whittaker gl–module 11

6. The structure of the Whittaker module M1(λ,µ) as a gl–module 147. Whittaker vectors in M1(λ,µ) 168. Bosonic realisations of M1(λ,µ) and L(d,λ,µ) 188.1. Realisation of M1(λ,µ) from Π(0)–modules 18

2010 Mathematics Subject Classification. Primary 17B69; Secondary 17B20, 17B65.Key words and phrases. vertex algebra, Whittaker modules, W1+∞, typical modules .

1

2 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

8.2. Free field realisation of L(d,λ,µ): the case M1(λ,µ) is a Π(0)–module. 208.3. The general case and non-tensor product modules 209. Generalized Whittaker modules for g = gl(2ℓ,C) 219.1. Weyl algebra A2ℓ and its Whittaker modules. 219.2. Embedding of g into A2ℓ. 229.3. W (α,β) as a g–modules. 22Appendix: M1(λ,µ) as a module for the Heisenberg-Virasoro algebra 23References 25

1. Introduction

The representation theory of Weyl algebras provides an important tool in the construction

of modules for finite and infinite-dimensional Lie algebras, and vertex algebras. Let A denotethe Weyl algebra, and let M denote the associated Weyl vertex algebra (βγ–system) of rankone. In [4], we have proved that irreducible Whittaker modulesM1(λ,µ) for the Weyl vertexalgebra M remain irreducible when we restrict them to the Zn–invariant subalgebra MZn .In this paper, we consider a limit case and restrict these modules to a Z–invariant subalgebraM0. Surprisingly, we prove that the irreducible Weyl modules of Whittaker type are alwaysreducible as M0–modules

However, the analysis of these reducible modules leads to many interesting observationsand new constructions.

First, we recall that the Z–invariant subalgebra M0 of the rank one Weyl vertex algebraM is isomorphic to the simple vertex algebra W1+∞,c at central charge c = −1 (cf. [14],[19]).Thus every Whittaker module for the Weyl vertex algebra is a module for W1+∞,c=−1.On the other hand, the representation theory of the vertex algebra W1+∞,c is deeply con-

nected with the representation theory of the Lie algebra gl, which is a central extensionof the Lie algebra of infinite matrices (cf. [13], [14]). More precisely, the universal vertex

algebra Wc1+∞ is the vacuum module of central charge c for the Lie algebra D, which is the

central extension of the Lie algebra of complex regular differential operators on C∗. More-

over, there exists a homomorphism Φ0 : D → gl, defined by (4.2); hence each gl–module

of central charge c is a D–module of the same central charge. If the resulting module isrestricted, we get a Wc

1+∞–module. Using the Weyl (vertex) algebras, one can construct

gl–modules which become modules for the simple vertex algebra W1+∞,c. Given the Lie

homomorphism gl → A defined by

Ei,j 7→: a(−i)a∗(j) :, C 7→ −1,

every restricted A–module has the structure of a gl–module, and one shows that they arealso W1+∞,c=−1–modules.

Similarly as in the case of the quasi-finite modules from [14], we show in Theorem 5.2that the structure of the module M1(λ,µ) as a W1+∞,c=−1–module is equivalent to the

structure of the same module as a gl–module.

WHITTAKER MODULES FOR gl AND W1+∞ 3

As far as we have noticed, Whittaker modules for the Lie algebra gl and for the vertexalgebra W1+∞ haven’t been investigated in the literature yet.

Hence our main task is to analyze M1(λ,µ) as a gl–module. This will directly imply thestructure of M1(λ,µ) as a W1+∞–module. We prove:

• M1(λ,µ) is a cyclic gl–module of Whittaker type for the Whittaker pair (gl, p),

where p is a certain commutative subalgebra of upper triangular matrices of gl.

Thus, M1(λ,µ) is not a classical Whittaker module for gl, but a generalized Whit-taker module in the sense of [7].

• In order to see that M1(λ,µ) is not irreducible, we use the Casimir element I =∑iEi,i of U(gl). We show that I acts non-trivially on M1(λ,µ).

• We prove that C[I]wλ,µ provides all Whittaker vectors in M1(λ,µ) and that forevery d ∈ C, the quotient

L(d,λ,µ) =M1(λ,µ)

〈(I − d)C[I]wλ,µ〉

is simple both as a gl–module and as W1+∞,c=−1–module.

Free field realization of L(d,λ,µ). Note that modules L(d,λ,µ) are constructed asquotients of Whittaker Weyl modules. A natural question is to provide an explicit realizationof these simple modules.

There exist another realization of Whittaker W1+∞,c–modules which are based on the factthat W1+∞,c can be embedded into Heisenberg vertex algebra for integral central charges.Let Mn(1) denotes the Heisenberg vertex algebra generated by n Heisenberg fields.

• W1+∞,c=1 is isomorphic to M1(1).• W1+∞,c=n is isomorphic to the principal W-algebra W1(gl(n), fprinc) which is avertex subalgebra of Mn(1). (cf. [10]).

• W1+∞,c=−1 is isomorphic to the tensor product M(2) ⊗M1(1), where M(2) is thesinglet vertex algebra at central charge −2 (isomorphic to the simpleW(2, 3)-algebraat c = −2) and M1(1) is rank one Heisenberg vertex algebra (cf. [19]). In particular,W1+∞,c=−1 is embedded into M2(1).

• In general, W1+∞,c=−n is embedded into M2n(1) (cf. [1]).

To the best of our knowledge, the following problem is still unsolved:

Problem 1.1. For embedding W1+∞,c → Mn(1) mentioned above, identify WhittakerMn(1)–modules as Whittaker W1+∞,c–module.

We will be focused on the case c = −1. Then W1+∞,c=−1 is a subalgebra of M2(1)and every Whittaker M2(1)–module is naturally Whittaker W1+∞,c=−1–module. Then byusing the isomorphism W1+∞,c=−1

∼= M(2) ⊗M1(1) one can realize a family of Whittakermodules as the tensor product modules

Z1 ⊗ Z2(1.1)

where Z1 is an irreducible Whittaker M(2)–module and Z2 is an irreducible WhittakerM1(1)–module. K. Tanabe showed in [18] that every Whittaker M(p)–module is obtainedfrom Whittaker modules for the Heisenberg vertex algebra. Therefore, we have a family

4 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

of irreducible W1+∞,c=−1–modules obtained using free-field realization from the WhittakerM2(1)–modules.

Question 1.2. Can we realize L(d,λ,µ) as irreducible Whittaker M2(1)–modules? This isequivalent to the question can we represent L(d,λ,µ) as the tensor product modules (1.1)?

In Section 8 we find a direct realization of L(d,λ,µ) asM2(1)–modules in the special caseswhen M1(λ,µ) has the structure of a module for the vertex algebra Π(0) (cf. Proposition8.2). Then the Casimir element I can be seen as an element of the vertex algebra Π(0).

But in general L(d,λ,µ) does not have form (1.1). The reason is that L(d,λ,µ) containsvectors which are not locally finite for the action the element J0(k), for k > 0, which is anecessary condition for a W1+∞,c=−1–module to have the form (1.1). Therefore modulesL(d,λ,µ) are irreducible modules of a completely new type.

We can slightly generalize these examples by applying the spectral-flow authomorphismρs of the Weyl algebras, which induces the spectral flow automorphism ρs of the Lie algebra

gl.Let us summerize:

Theorem 1.3. The vertex algebra W1+∞,c=−1 has the following two families of irreduciblemodules of Whittaker types:

(1) Tensor product modules Z1 ⊗ Z2, where Z1 is a Whittaker modules for the singletalgebra M(2) (cf. [18]) and Z2 is a Whittaker module for the Heisenberg vertexalgebra M1(1).

(2) Modules ρs(L(d,λ,µ)) which are irreducible quotients of the Weyl Whittaker mod-ules ρs(M1(λ,µ)).

Moreover, the following statements are equivalent (cf. Propositions 8.2, 8.3):

(a) Module in (2) is of type (1);(b) M1(λ,µ) has the structure of a Π(0)–module;(c) λ = (λ0, 0, · · · ), λ0 6= 0.

Typical W1+∞,c–modules. Weak modules for the vertex operator algebra W1+∞,c can benaturally divided in two classes: typical and atypical. We say that a weak W1+∞,c–moduleis typical if it is irreducible as a module for the Heisenberg-Virasoro vertex subalgebra ofW1+∞,c, which we denote by LHV ir

c . In the case c = −1, one can construct typical/atypicalhighest weight W1+∞,c–modules by using typical/atypical modules for the singlet vertexalgebra (cf. [2, 6, 19, 8]).

Tensor product modules in Theorem 1.3(1) are typical since Whittaker M(2)–modulesare realized on irreducible, Whittaker Virasoro modules (cf. [18]). In Theorem 9.4 inAppendix we prove typicality of our new series of modules:

Theorem 1.4. Modules L(d,λ,µ)) are typical W1+∞,c=−1–modules.

A connection with Whittaker gl(2ℓ)–modules. The study of Whittaker modulesM1(λ,µ)

as gl–modules is motivated by [4] and the orbifold theory of vertex operator algebras. As

a result we get a nice family of Whittaker modules for the Lie algebra gl and describe acomplete set of Whittaker vectors. Since one can embed finite-dimensional Lie algebra gl(n)

into gl, it is a natural question what is gl(n)–version of the Whittaker modules M1(λ,µ). It

WHITTAKER MODULES FOR gl AND W1+∞ 5

turns out that the theoretical background for these modules was given in [7] in the context ofgeneralized Whittaker modules. But the paper [7] only presented a construction of universalgeneralized gl(n)–modules, and it seems the corresponding simple quotients haven’t beeninvestigated before our paper. Because of simplicity we consider here the gl(2ℓ)–modulesW (α,β), where α,β ∈ Cℓ. which are quotients of universal generalized Whittaker modulesfrom [7]. We present this construction in Section 9. We describe the structure of W (α,β)and obtained a complete family of Whittaker vectors. We prove that if α,β 6= 0, then

L(d,α,β) =W (α,β)

〈(I − d)C[I]wα,β〉

is a simple generalized Whittaker gl(2ℓ)–module.

Acknowledgements. We would like to thank C. H. Lam, N. Yu, A. Romanov and K.Zhao on discussion about Whittaker modules.

The authors are partially supported by the QuantiXLie Centre of Excellence, a projectcofinanced by the Croatian Government and European Union through the European Re-gional Development Fund - the Competitiveness and Cohesion Operational Programme(KK.01.1.1.01.0004).

2. Preliminaries

2.1. Classical Whittaker modules. For the structural theory of Whittaker modules forWhittaker pairs see [7].

Assume that g is a Lie algebra with triangular decomposition g = n− ⊕ h⊕ n+.Let W be a g–module. Vector v ∈ W is called a Whittaker vector if there is a Lie

functional λ : n+ → C such that xv = λ(x)v for all x ∈ n+.Let λ : n+ → C be a Lie functional. Let Cvλ be a 1–dimensional n+–module such that

xvλ = λ(x)vλ for all x ∈ n+. Then

Mλ = U(g)⊗U(n+) Cvλ,

is a U(g)–module, which is called the universal (or standard) Whittaker module.

2.2. Generalized Whittaker modules. Let g be a Lie algebra, and n any nilpotentsubalgebra of g. Whittaker modules with respect to n are sometimes called generalizedWhittaker modules, or Whittaker modules with respect to the pair (g, n). For example, wecan take n to be any commutative subalgebra of n+.

This makes this situation more general to the previous one where the nilpotent subalgebrais exactly n+.

Let W be a g–module. Vector v ∈ W is called Whittaker vector for the pair (g, n) if thereis a Lie functional λ : n → C such that xv = λ(x)v for all x ∈ n.

Let λ : n → C be a Lie functional. Let Cvλ be a 1–dimensional n–module such thatxvλ = λ(x)vλ for all x ∈ n. Then

Mλ = U(g)⊗U(n) Cvλ,

is a U(g)–module, which is called the universal Whittaker module for the pair (g, n).

6 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

Such Whittaker modules and corresponding Whittaker vectors are sometimes called thegeneralized Whittaker modules/Whittaker vectors.

2.3. Example: g = gl(2ℓ,C). In this Subsection we take g = gl(2ℓ,C). Take the usualbasis ei,j, i, j = 1, . . . , n of g.

Recall the triangular decomposition: g = n− ⊕ h⊕ n+. Here

n+ = spanC{ei,j | i < j},

n− = spanC{ei,j | i > j},

h = spanC{ei,i | i = 1, . . . , 2ℓ}.

Note that n+ is not commutative. For example [e1,2, e2,3] = e1,3.Whittaker modules with respect to this triangular decomposition are called classical

Whittaker modules.But we can take the following subalgebra n ⊂ n+.

n = spanC{ei,j+ℓ |i, j = 1, . . . , ℓ}.

Then n is a commutative Lie algebra.Whittaker modules with respect to the pair (g, n) are called the generalized Whittaker

modules. Let us describe these modules. Take λ ∈ n∗. Consider the universal Whittakermodule

Mλ = U(g)⊗U(n) Cvλ.

Now we have two natural questions:

• Is Mλ irreducible?• If Mλ is reducible, describe simple quotients of Mλ.

These modules appeared in [7, Example 23]. Quite surprisingly, the irreducibility analysishas not been presented neither in [7] or at any other recent work. We shall see later thatMλ is never irreducible, and describe its simple quotients.

2.4. Whittaker modules for g = gl. The basis is given by (cf. [13, 14]):

C, Ei,j, i, j ∈ Z.

The triangular decomposition: g = n− ⊕ h⊕ n+. Here

n+ = spanC{Ei,j | i < j},

n− = spanC{Ei,j | i > j},

h = CC ⊕ spanC{Ei,i | i ∈ Z}.

Let

p = spanC{E−i−1,j |i, j ∈ Z≥0}.

Then p is a commutative subalgebra of n+.As before we call the Whittaker modules for the Whittaker pair (g, n+) the classical

Whittaker gl–modules, and Whittaker modules for the pair (g, p) the generalized Whittaker

gl–modules.

In what follows we shall describe a family of simple, generalized Whittaker gl–modules.

WHITTAKER MODULES FOR gl AND W1+∞ 7

3. Weyl vertex algebra and its Whittaker modules

We now recall the notion of Weyl vertex algebra as this will be our main object of study.

To define this vertex algebra, first we remind the reader of the notion of Weyl algebra ALet L be the infinite-dimensional Lie algebra with generators

K,a(n), a∗(n), n ∈ Z,

such that K is in the center where and the only notrivial relations are

[a(n), a∗(m)] = δn+m,0K, n,m ∈ Z.

The Weyl algebra A is:

A =U(L)

〈K − 1〉,

where 〈K − 1〉 is the two sided ideal generated by K − 1. So, in A we have K = 1.To construct the Weyl vertex algebra, we need a vector space, so we choose the simple

Weyl algebra module M generated by a cyclic vector 1 such that

a(n)1 = a∗(n+ 1)1 = 0 (n ≥ 0),

that is, M = C[a(−n), a∗(−m) | n > 0, m ≥ 0].Now, by the generating fields theorem, there is a unique vertex algebra (M,Y,1) where

the vertex operator is given by

Y : M → End(M)[[z, z−1]]

such that

Y (a(−1)1, z) = a(z), Y (a∗(0)1, z) = a∗(z),

a(z) =∑

n∈Z

a(n)z−n−1, a∗(z) =∑

n∈Z

a∗(n)z−n.

We choose the following conformal vector of central charge c = −1 (cf. [14]):

ω =1

2(a(−1)a∗(−1)− a(−2)a∗(0))1.

Then (M,Y,1, ω) has the structure of a 12Z≥0–graded vertex operator algebra. Weak and

ordinary modules for (M,Y,1, ω) can be defined as in the case of Z–graded vertex operatoralgebras.

Following [4], we define the Whittaker module for A to be the quotient

M1(λ,µ) = A/I,

where λ = (λ0, . . . , λn), µ = (µ1, . . . , µm) and I is the left ideal

I =⟨a(0) − λ0, . . . , a(n)− λn, a

∗(1) − µ1, . . . , a∗(m)− µm, a(n + 1), . . . , a∗(n+ 1), . . .

⟩.

Let n be the subalgebra of L generated by a(n), a∗(n + 1), n ∈ Z≥0. Then n is commu-tative, and therefore nilpotent subalgebra of L.

Proposition 3.1. [4] We have:

8 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

(1) M1(λ,µ) is a standard Whittaker module for the Whittaker pair (L, n) of level K = 1with the Whittaker function Λ = λ,µ : n → C:

Λ(a(i)) = λi (i = 0, . . . , n), Λ(a(i)) = 0 (i > n)

Λ(a∗(i)) = µi (i = 1, . . . ,m), Λ(a∗(i)) = 0 (i > m).

(2) M1(λ,µ) is an irreducible A–module.(3) M1(λ,µ) is an irreducible weak module for the Weyl vertex operator algebra M .

The Whittaker vector will be denoted by wλ,µ.

For each s ∈ Z, the Weyl algebra A contains the following automorphism ρs:

ρs(a(n)) = a(n+ s), ρs(a∗(n)) = a∗(n− s), (n ∈ Z).

For any A–module N , let ρs(N) denote the A–module ρs(N) such that ρs(N) = N as avector space and

x.v = ρs(x)v, x ∈ A, v ∈ N.

Clearly, N is irreducible A–module if and only if ρs(N) is irreducible A–module.

Let n(s) denote the commutative subalgebra of L generated by a(n − s), a∗(n + 1 + s),n ∈ Z≥0.

Proposition 3.2. For each s ∈ Z we have:(1) ρs(M1(λ,µ)) is a standard Whittaker module for the Whittaker pair (L, n(s)) of levelK = 1 with the Whittaker function Λ(s) : n → C:

Λ(s)(a(i − s)) = λi (i = 0, . . . , n), Λ(s)(a(i− s)) = 0 (i > n)

Λ(s)(a∗(i+ s)) = µi (i = 1, . . . ,m), Λ(s)(a∗(i+ s)) = 0 (i > m).

(2) ρs(M1(λ,µ)) is an irreducible A–module.(3) ρs(M1(λ,µ)) is an irreducible weak module for the Weyl vertex operator algebra M .

Let ζp = e2πi/p be p-th root of unity. Let gp be the automorphism of the vertex operatoralgebra M which is uniquely determined by the following automorphism of the Weyl algebra

A:a(n) 7→ ζpa(n), a∗(n) 7→ ζ−1

p a∗(n) (n ∈ Z).

Then gp is the automorphism of M of order p.The following result was proved in [4] in the case s = 0 and the proof for arbitrary s ∈ Z

is completely analogous.

Theorem 3.3. [4] Assume that Λ = (λ,µ) 6= 0 and s ∈ Z. Then ρs(M1(λ,µ)) is an

irreducible weak module for the orbifold subalgebra MZp = M 〈gp〉, for each p ≥ 1.

Let us define an operator J0 :=: aa∗ :. Then the components of the field

Y (J0, z) =∑

n∈Z

J0(n)z−n−1

satisfies the commutation relations for the Heisenberg algebra at level −1. We have thefollowing subalgebra of M :

M0 = KerMJ0(0).

WHITTAKER MODULES FOR gl AND W1+∞ 9

Let ζ ∈ C, |ζ| = 1, which is not a root of unity. Let g be the automorphism of thevertex operator algebra M uniquely determined by the following automorphism of the Weyl

algebra A:

a(n) 7→ ζa(n), a∗(n) 7→ ζ−1a∗(n) (n ∈ Z).

Then g is the automorphism of M of infinite order, and the group G = 〈g〉 is isomorphic toZ. Clearly, we have

M0 ∼= MG.

Note also that

M0 =

∞⋂

p=1

MZp , M ⊃ MZ2 ⊃ · · ·MZp ⊃ · · · ⊃ M0.

Since M1(λ,µ) is an irreducible MZp–module for each p ≥ 1, it is natural to ask ifM1(λ,µ) is irreducible also as an M0–module. However, in this paper we prove thatM1(λ,µ) is a reducible and indecomposable M0–module.

Let us finish this section by describing the connections between ρs(M1(λ,µ)) andM1(λ,µ)as modules for vertex subalgebras invariant for J0(0). As M–modules these modules areconstructed by applying the automorphism ρs which has a nice description in terms of gen-

erators of A. But, the automorphism ρs does also have sense as automorphism of vertexJ0(0)–invariant subalgebras, which can be explain by using H. Li’s ∆–operator.

First we notice that as M–modules (see [5] for a proof):

(ρs(M1(λ,µ)), Ys(·, z)) = (M1(λ,µ), Y (∆(−sJ0, z)·, z))

where

∆(h, z) = zh(0) exp

(∞∑

n=1

h(n)

−n(−z)−n

).

In particular, the action of J0(z) on ρs(M1(λ,µ)) is given by

J0s (z) = J0(z) +

(−z)−1

−1Y (−sJ0(1)J0, z) = J0(z) + sz−1Id.

By restricting this realization on vertex subalgebras of M which are J0(0)–invariant weget:

Proposition 3.4. Let U be one of the subalgebras MZp or M0. As an U–module ρs(M1(λ,µ))is obtained from M1(λ,µ) as follows:

(ρs(M1(λ,µ)), Ys(v, z)) = (M1(λ,µ), Y (∆(−sJ0, z)v, z))

where v ∈ U .

Remark 1. In the case U = M0, the previous proposition shows that if we describe thestructure of M1(λ,µ) as a U–module (i.e., description of submodules, irreducible quotients)we will automatically describe the structure of ρs(M1(λ,µ)) as U–module.

.

10 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

4. W1+∞-algebra at central charge c = −1 and its Whittaker modules

A peculiarity of the orbifold M0 is that it has two additional important realizations whichwe plan to explore in our paper:

• M0 is isomorphic to the vertex algebra W1+∞-algebra at central charge c = −1.

• M0 is isomorphic to the simple module for the Lie algebra gl, which is the centralextension of the Lie algebra of infinite matrices.

4.1. The W1+∞-algebra approach. The universal vertex algebra Wc1+∞ is generated by

the fields

Jk(z) =∑

n∈Z

Jk(n)z−n−k−1 (k ∈ Z≥0),

whose components satisfy the commutation relations for the Lie algebra D at central chargec, which is a central extension of the Lie algebra of complex regular differential operatorson C∗ (cf. [10], [14]). It has a simple quotient, which we denote by W1+∞,c.

The components of the fields J0(z), J1(z) satisfies the commutation relation for theHeisenberg Virasoro Lie algebra H generated by J0(r), J1(r), C1, C2 with the commuta-tion relations (r, s ∈ Z):

[J0(r), J0(s)] = rδr+s,0C2,

[J1(r), J0(s)] = −sJ1(r + s),

[J1(r), J1(s)] = (r − s)J1(r + s) +r3 − r

12δr+s,0C1,

with central elements C1 and C2. On each weak W1+∞,c–module the actions of the centralelements are given by C1 = −2c, C2 = c. Let LHV ir

c be the Heisenberg-Virasoro vertexsubalgebra of W1+∞,c generated by J0(z), J1(z).

Definition 4.1. An irreducible weak W1+∞,c–module M is called typical (resp. atypical) if

it is irreducible (resp. reducible) as a LHV irc –module.

It was proved by V. Kac and A. Radul in [14] that M0 ∼= W1+∞,c for c = −1. As aconsequence, we have that M0 is generated by the fields

Jk(z) = Y (a∗(−k)a, z) =:(∂kz a

∗(z))a(z) :=

∑

n∈Z

Jk(n)z−n−k−1 (k ∈ Z≥0).

Remark 2. One can show that modules obtained in the decomposition of M as a M0–moduleare atypical (cf. [19]), but the modules constructed from the irreducible relaxed modules in[5] are typical highest weight W1+∞,c=−1–modules.

Our Whittaker modules for the Weyl vertex algebra are automatically Whittaker weakmodules for W1+∞,c=−1.

4.2. Approach using the Lie algebra gl. Define the generating function

E(z, w) =: a(z)a∗(w) :=∑

i,j∈Z

Ei,jzi−1w−j .

WHITTAKER MODULES FOR gl AND W1+∞ 11

In other words, the operators Ei,j are defined as

Ei,j =: a(−i)a∗(j) :(4.1)

These operators endow M with the structure of a gl–module at central charge K = −1 (see

formula (2.7) in [14] for commutation relations for gl).We have (cf. [14]):

[Ei,j, a(−m)] = δj,ma(−i), [Ei,j, a∗(m)] = −δi,ma∗(j).

4.3. Connection between gl–modules and W1+∞–modules. In [10], [14] the authors

demonstrated how one can construct W1+∞–modules from gl–modules. Strictly speaking,their approach was used only for quasi-finite modules, which are necessarily weight modules.Therefore, they formulated their results for a category of modules which does not includeWhittaker modules. However, we extend their approach to a broader category which alsoincludes Whittaker modules.

Following [10], we have a homomorphism of Lie algebras Φ0 : D → gl, determined by

Φ0(tmf(D)) =

∑

j∈Z

f(−j)Ej−m,j, Φ0(C) = K,(4.2)

where f ∈ C[x], D = t∂t. This homomorphism enables us to consider gl–modules as

D–modules, and therefore as modules for the vertex algebra Wc1+∞.

It seems that the representation theory of gl is easier for analysis than the representationtheory of Wc

1+∞. Nonetheless, a problem is in the fact that Φ0 is not surjective homomor-

phism. Hence the restriction of irreducible gl–module to D (and therefore to Wc1+∞) need

not be irreducible. However, for a family of quasi-finite modules, V. Kac and A. Radul [13]showed that the image of Φ0 is dense in a certain way which we explain latter (cf. [13,

Proposition 4.3]). So for quasi-finite modules, irreducible gl–modules are also irreducibleWc

1+∞–modules. Our goal is to show that we can replace quasi-finite modules with certain

Whittaker modules, so that irreducible gl–modules will become irreducible for Wc1+∞.

5. M1(λ,µ) as a Whittaker gl–module

Proposition 5.1. Assume that λ 6= 0 and µ 6= 0. Then M1(λ,µ) is a Whittaker gl–module

for the pair p ⊂ gl at central charge K = −1, generated by the Whittaker vector wλ,µ.

Proof. Let us prove that wλ,µ is indeed cyclic. As a vector space, M1(λ,µ) ∼= M , so itsbasis elements are of the form

w = a(−n1) · · · a(−nr)a∗(−m1) · · · a

∗(−ms)wλ,µ.

We will now use double induction on the length of a vector in M1(λ,µ). First inductivehypothesis is the following:

Every element of the form w = a(−n1) · · · a(−nr)wλ,µ is an element of U(gl).wλ,µ.

12 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

Let us first prove the initial case. Let a(−n)wλ,µ be an element of M1(λ,µ). Sinceµ 6= 0, there is an index j0 such that µj0 6= 0, and a∗(j0)wλ,µ = µjwλ,µ. Therefore, wehave

1

µj0

Ei,j0wλ,µ = a(−n)wλ,µ.

Now, let us assume that for any vector of lenght at most r, the inductive hypothesisholds. Let us take a vector of the form

v = a(−n1)a(−n2) · · · a(−nr+1)wλ,µ.

We have that

1

µr+1j0

En1,j0En2,j0 · · ·Enr+1,j0wλ,µ

=1

µr+1j0

: a(−n1)a∗(j0) : a(−n2)a

∗(j0) : . . . : a(−nr+1)a∗(j0) : wλ,µ

=1

µr+1j0

a(−n1)a(−n2) · · · a(−nr+1) a∗(j0)a

∗(j0) · · · a∗(j0)︸ ︷︷ ︸

r+1factors

wλ,µ

+ C11

µr+1j0

a(−n1) · · · a(−nk) · · · a(−nr+1) a∗(j0) · · · a

∗(j0)︸ ︷︷ ︸rfactors

wλ,µ

+ . . .+ Cr+1wλ,µ,

=1

µr+1j0

µr+1j0

v +K,

where Ci are constants that can be zero, a(−n) means that this element is left out, and Kis a (possibly zero) sum of vectors of length at most r. Therefore, we have that

v =1

µr+1j0

En1,j0En2,j0 · · ·Enr+1,j0wλ,µ −K,

and by the induction hypothesis, we have that v ∈ U(gl).wλ,µ.The second induction hypothesis is that every vector of the form

a∗(−m1) · · · a∗(−ms)a(−n1) · · · a(−nr)wλ,µ

is in U(gl).wλ,µ.Let us first prove the initial case. Let v = a∗(−m)a(−n1)a(−n2) · · · a(−nr)wλ,µ, where

r is any non-negative integer. Since λ 6= 0, there is an index λi0 6= 0. Therefore,

1

λi0

E−i0,−ma(n1) · · · a(−nr)wλ,µ

=1

λi0

: a(i0)a∗(−m) : a(n1) · · · a(−nr)wλ,µ

= v.

WHITTAKER MODULES FOR gl AND W1+∞ 13

Now, let us assume that for any vector with at most s factors a∗() the inductive hypothesisholds. Let us take a vector of the form

v = a∗(−m1) · · · a∗(−ms+1)a(−n1)a(−n2) · · · a(−nr)wλ,µ.

We have that1

λs+1i0

E−i0,−m1E−i0,−m2 · · ·E−i0,−ms+1a(−n1)a(−n2) · · · a(−nr)wλ,µ

=1

λs+1i0

: a(i0)a∗(−m1) : a(i0)a

∗(−m2) : . . . : a(i0)a∗(−ms+1) : a(−n1) · · · a(−nr)wλ,µ

=1

λs+1i0

a∗(−m1)a∗(−m2) · · · a

∗(−ms+1)a(−n1) · · · a(−nr) a(i0)a(i0) · · · a(i0)︸ ︷︷ ︸s+1factors

wλ,µ

+D11

λs+1i0

a∗(−m1) · · · a∗(−mk) · · · a∗(−ms+1)a(−n1) · · · a(−nr) a(i0) · · · a(i0)︸ ︷︷ ︸

sfactors

wλ,µ

+ . . .+Ds+1wλ,µ,

=1

λs+1i0

λs+1i0

v + L,

where Di are constants that can be zero, a∗(−m) means that this element is left out, andL is a (possibly zero) sum of vectors of length at most s. Therefore, we have that

v =1

λs+1i0

E−i0,−m1E−i0,−m2 · · ·E−i0,−ms+1a(−n1)a(−n2) · · · a(−nr)wλ,µ − L,

and by the induction hypothesis, we have that v ∈ U(gl).wλ,µ.This completes the proof that the Whittaker vector wλ,µ is cyclic. �

Next, using a connection between gl–modules and W1+∞–modules we have:

Theorem 5.2. Assume that λ 6= 0 and µ 6= 0. Then M1(λ,µ) is a Whittaker W1+∞,c=−1–module generated by the cyclic vector wλ,µ. In particular, M1(λ,µ) is a cyclic module forthe orbifold vertex algebra M0.

Proof. By Proposition 5.1 we have that M1(λ,µ) is a cyclic gl–module and we want to

prove it is a cyclic D–module. Take a homomorphism Φ0 : D → gl. Although M1(λ,µ) isnot a quasi-finite weight module, we will show that one can still apply [13, Proposition 4.3].We use the following arguments:

• Let M = M1(λ,µ). Recall that M ∼= M as a vector spaces.• Take the Virasoro vector ω = 1

2(a(−1)a∗(−1) − a(−2)a∗(0))1 of central charge

c = −1 and define L(n) = ωn+1. Then L(0) defines a 12Z≥0 gradation on M :

M =⊕

m∈12Z≥0

Mm

such that v ∈ Mm ⇐⇒ L(0)v = mv.

14 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

• Define M(m) =⋃

k≤mMk.

• Since dimMk < ∞ for all k ∈ 12Z≥0, we have that dimM(m) < ∞ for all m ∈ 1

2Z≥0.• Then by applying the homomorphism Φ0 on M1(λ,µ) we get:

Φ0(tmf(D)) =

∑

j∈Z

f(−j)Ej−m,j , Φ0(C) = −1(5.1)

=∑

j∈Z

f(−j) : a(m− j)a∗(j) :

• For v ∈ M(m), we have that Φ0(tmf(D))v ∈ M(m + N), where N is determined

by Whittaker function, has the property

λi = µi = 0 ∀i > N,

and can be chosen independently on m.• Therefore Φ0(t

mf(D)) defines a homomorphism

(∗) C[x] → Hom(M(m),M(m +N)).

• Since dimM(m) < ∞ and dimM(m +N) < ∞, the proof of [13, Proposition 4.3]gives that (*) is a continuous map. Therefore, it can be uniquely extended to ahomomorphism

(∗) O → Hom(M(m),M(m +N)),

where O denotes the algebra of holomorphic functions on C.• Since for each j ∈ Z there is a holomorphic function g such that g(ℓ) = δℓ,j, ℓ ∈ Z,we conclude that each Ei,j is in the image of map (5.1).

• Since M is a Whittaker gl–module generated by wλ,µ, M is also generated by wλ,µ

as a D–module.

The proof follows. �

6. The structure of the Whittaker module M1(λ,µ) as a gl–module

We have proved in Proposition 5.1 that M1(λ,µ) is a cyclic gl–module. Let us provethat it is not irreducible. We will use the following Casimir element.

Define

I :=∑

j∈Z

Ej,j.

As far we can see, I is introduced in [12] for a slightly different category of modules.

However, it is well defined on the family of Whittaker gl–modules which we considered.

Lemma 6.1. On any weak M–module M we have that I = J0(0). In particular:

(1) I ∈ End(M1(λ,µ)).

(2) The action of I commutes with the action of gl on M1(λ,µ).

WHITTAKER MODULES FOR gl AND W1+∞ 15

Proof. On each weak M–module the operator J0(n) =∑

j∈Z : a(j + n)a(−j) : is welldefined. In particular, we have:

J0(0) =∑

j∈Z

: a(−j)a∗(j) := I.

In particular this implies that I is well defined operator acting on M1(λ,µ). Thus (1) holds.

For the proof of assertion (2) we use commutation relations for gl:

[Ei,j, Es,t] = δj,sEi,t − δi,tEs,j − CΦ(Ei,j, Es,t).

We calculate:

[Ei,j, I]v =

(∑

k∈Z

[Ei,j , Ek,k]

)v

= (Ei,j − Ei,j) v

= 0.

Therefore, our second assertion also holds.�

Lemma 6.2. (1) For every n ∈ Z≥1, Inwλ,µ is an non-trivial Whittaker vector inM1(λ,µ) .

(2) For any S ⊂ C[I]wλ,µ, let 〈S〉 be the submodule generated by Whittaker vectorsfrom S. Then 〈(I − d)C[I]wλ,µ〉 is a proper submodule of M1(λ,µ) for each d ∈ C

.

We have proved the following result.

Theorem 6.3. M1(λ,µ) is a reducible, cyclic gl–module.

The automorphism ρs of A induces the following automorphism of gl:

ρs : Ei,j =: a(−i)a∗(j) 7→ Ei−s,j−s =: a(−i+ s)a∗(j − s) : .

So this means that as gl–module ρs(M1(λ,µ)) is obtained from M1(λ,µ) by applying theautomorphism ρs. As a consequence we gat

Corollary 6.4. We have:

(1) ρs(M1(λ,µ)) is a reducible, cyclic gl–module.(2) Let L be any irreducible quotient of M1(λ,µ). Then ρs(L) is a irreducible quotient ofρs(M1(λ,µ)).

Let L be any irreducible quotient of M1(λ,µ). (We will see bellow that it is not unique).Two very important problems arise:

(A) Find an explicit realization of L if possible.(B) Determine the complete set of Whittaker vectors in M1(λ,µ) which generate the

maximal submodule of M1(λ,µ).

In what follows we will solve both problems.

16 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

7. Whittaker vectors in M1(λ,µ)

In this section we describe the complete set of Whittaker vectors in M1(λ,µ) and deter-mine its simple quotients. In particular, we completely solve the problem (B).

Let M = M1(λ,µ).As before, we choose N ∈ Z≥0 such that a(n)wλ,µ = a∗(n)wλ,µ = 0 for n ≥ N . Then

v = Iwλ,µ = (: a(0)a∗(0) : + · · ·+ : a(N)a∗(−N) :)wλ,µ

+(: a(−1)a∗(1) : + · · ·+ : a(−N)a∗(N) :)wλ,µ

=N∑

k=0

λka∗(−k)wλ,µ +

N∑

k=1

µka(−k)wλ,µ(7.1)

is a new Whittaker vector. Now we choose i0 and j0 such that

λi0 · µj0 6= 0.

Lemma 7.1. For n, k ≥ 0 we have

a(n)Ikwλ,µ = λn(I + 1)kwλ,µ.

a∗(n+ 1)Ikwλ,µ = µn+1(I − 1)kwλ,µ.

Proof. By direct calculation we have

[a(n), I] =∑

j∈Z

[a(n), Ej,j ] = [a(n), a∗(−n)a(n)] = a(n).

This implies that

a(n)Iwλ,µ = λnIwλ,µ + [a(n), I]wλ,µ = λn(Iwλ,µ +wλ,µ) = λn(I + 1)wλ,µ.

a(n)I2wλ,µ = λn(I2 + 2I + 1)wλ,µ = λn(I + 1)2wλ,µ

Assume that a(n)Ikwλ,µ ∈ C[I]wλ,µ. Then

a(n)Ik+1wλ,µ = I(a(n)Ikwλ,µ) + a(n)Ikwλ,µ = λn(I(I + 1)k + (I + 1)k)wλ,µ.

= (I + 1)k+1wλ,µ.

Now first claim holds by the induction. The proof of the second claim is analogous. �

Consider now some examples

E−i0,n1a(−n1)Ikwλ,µ = [E−i0,n1 , a(−n1)]

kIkwλ,µ + a(−n1)E−i0,n1Ikwλ,µ

= −a(i0)Ikwλ,µ + a(−n1)I

kE−i0,n1wλ,µ

= −λi0(I + 1)kwλ,µ + λi0µn1a(−n1)Ikwλ,µ

Em1,j0a∗(−m1)I

kwλ,µ = [Em1,j0 , a∗(−m1)]I

kwλ,µ + a∗(−m1)Em1,j0Ikwλ,µ

= a∗(j0)Ikwλ,µ + a∗(−m1)I

kEm1,j0wλ,µ

= µj0(I − 1)kwλ,µ + λm1µj0a∗(−m1)I

kwλ,µ

WHITTAKER MODULES FOR gl AND W1+∞ 17

This implies:

Em1,j0a∗(−m1)I

kwλ,µ =1

µj0

(Em1,j0 − λm1µj0)a∗(−n1)I

kwλ,µ = (I − 1)kwλ,µ,

andIn this way we get:

Lemma 7.2. Let p ∈ Z≥0.(1) Let

Φ ∈ C[a(−n− 1), a∗(−m) |n,m ∈ Z≥0,m 6= i0}.

Then

E−i0,p ΦIkwλ,µ =

(∂

∂a(−p)Φ

)(I + 1)kwλ,µ.

(2) Let

Φ∗ ∈ C[a∗(−m) |m ∈ Z≥0,m 6= i0}.

Then

Ep,j0 Φ∗Ikwλ,µ =

(∂

∂a∗(−p)Φ

)(I − 1)kwλ,µ.

Proposition 7.3. Assume that v is a Whittaker vector in M. Then v ∈ C[I]wλ,µ.

Proof. First we see that as a vector space:

M ∼= C[a(−n− 1), a∗(−m) |n,m ∈ Z≥0,m 6= i0} ⊗ C[I],

i.e., the basis of M consists of the vectors:

(∗) a(−n1) · · · a(−nr)a∗(−m1) · · · a

∗(−ms)Ikwλ,µ, ni ≥ 1,mi ≥ 0,mi 6= i0, k ≥ 0.

So we can assume that we have a Whittaker vector of the form

v =

k∑

j=0

ΦjIjwλ,µ(7.2)

for certain polynomials Φj ∈ C[a(−n− 1), a∗(−m) |n,m ∈ Z≥0,m 6= i0}. By using Lemma7.2 (1) we get

E−i0,p v =

k∑

j=0

(∂

∂a(−p)Φj

)(I + 1)jwλ,µ.

Since v is a Whittaker vector E−i0,p must act trivially on it. Since vectors (*) are linearlyindependent we conclude that

∂

∂a(−p)Φj = 0 ∀p ∈ Z>1, j = 0, . . . , k.

This implies that Φj = Φ∗j for certain Φ∗

j ∈ C[a∗(m) |m ∈ Z≥0,m 6= i0} and

v =k∑

j=0

Φ∗jI

jwλ,µ

18 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

Now using Lemma 7.2 (2) we get

Ep,j0v =

k∑

j=0

(∂

∂a∗(−p)Φ∗j

)(I − 1)jwλ,µ.

Since v is a Whittaker vector, we have Ep,j0v = 0, implying that ∂∂a∗(−p)Φ

∗j = 0 ∀p ≥ 0.

This implies that Φ∗j is a constant, and therefore v ∈ C[I]wλ,µ. The Claim holds. �

As a consequence, we get description of irreducible Whittaker modules.

Theorem 7.4. For each d ∈ C:

L(d,λ,µ) =M1(λ,µ)

〈(I − d)C[I]wλ,µ〉

is irreducible.

Proof. It suffices to prove that each vector v of the form (7.2), whose projection to L(d,λ,µ)is non-zero, is cyclic. By using exactly the same arguments as in the proof of Proposition

7.3 we see that wλ,µ ∈ U(gl).v. Since wλ,µ is cyclic vector in M1(λ,µ), we conclude thatv is cyclic. The proof follows. �

By Applying Corollary 6.4 on the irreducible gl–module L(d,λ,µ) we get:

Corollary 7.5. For any s ∈ Z, ρs(L(d,λ,µ)) is an irreducible quotient of ρs(M1(λ,µ)).

8. Bosonic realisations of M1(λ,µ) and L(d,λ,µ)

K. Tanabe in [18] showed that the Whittaker modules for the rank one Heisenberg vertexalgebra remain irreducible when we restrict them to the singlet vertex algebra introducedin [2]. Since W1+∞-algebra at central charge c = −1 is isomorphic to the tensor productW3,−2 ⊗M1(1), where W3,−2 is the singlet vertex algebra at central charge −2 and M1(1)is rank one Heisenberg vertex algebra (cf. [19]), W1+∞,c=−1 can be realized as a subalgebraof the rank two Heisenberg vertex algebra M2(1).

Using Tanabe’s result, one shows that irreducible Whittaker M2(1)–modules remain ir-reducible when we restrict them to the vertex algebra W1+∞,c=−1. But it is a hard problemto identify them with our Whittaker modules L(d,λ,µ) realised as quotients of Whittakermodules for the Weyl vertex algebra. In this article, we identify modules L(d,λ,µ) asM2(1)–modules in the cases when M1(λ,µ) has the structure of a Π(0)–module which willbe described Subsection 8.1. But in general, our Whittaker modules are different to thoseobtained from Tanabe paper.

8.1. Realisation of M1(λ,µ) from Π(0)–modules. Let VL = Mγ,δ(1) ⊗ C[L] be thelattice vertex algebra associated to the lattice L = Zγ + Zδ with products

〈γ, γ〉 = 〈δ, δ〉 = 0, 〈γ, δ〉 = 2,

where Mγ,δ(1) is the Heisenberg vertex algebra generated by γ(z), δ(z), and C[L] the groupalgebra of the lattice L.

WHITTAKER MODULES FOR gl AND W1+∞ 19

Consider the following subalgebra of VL:

Π(0) = Mγ,δ(1) ⊗ C[Zγ].

Let Mϕ(0) be the commutative vertex algebra generated by the commutative even fieldϕ(z) =

∑i∈Z ϕ(i)z

−i−1, so

Mϕ(0) = C[ϕ(−1), · · · ].

For χ = χ(z) =∑

n∈Z χiz−i−1 ∈ C((z)), let Lϕ(χ) = C1χ be the 1–dimensional Mϕ(0)–

module such that ϕ(i) ≡ χiId on Lϕ(χ).

Note that γ, δ = δ − 2ϕ, e±γ generate a subalgebra Π(0) of Mϕ(0) ⊗ Π(0) and that

Π(0) ∼= Π(0).

There is an embedding of vertex algebras Φ : M → Π(0) → Mϕ(0)⊗Π(0) given by

Φ(a) = eγ ,Φ(a∗) = −1

2(γ(−1) + δ(−1) − 2ϕ(−1))e−γ .

Let a−1 = e−γ . Define a(n) = eγn, a−1(n) = e−γn−2. For λ ∈ C, λ 6= 0, let Πλ be the Z≥0–

graded irreducible Whittaker Π(0)–module (cf. [3]), generated by the Whittaker vector vλsuch that

a(n)vλ = λδn,0vλ, a−1(n)vλ =1

λδn,0vλ (n ∈ Z≥0).

As vector spaces:

Πλ∼= C[γ(−n), δ(−n + 1) | n ∈ Z>0], (Πλ)top = C[δ(0)]vλ ∼= C[δ(0)].

Now consider module Lϕ(χ)⊗Πλ. Define wλ,χ = 1χ ⊗ vλ.Assume that µ = (µ1, µ2, · · · , µn)

χi = λµi, i = 1, . . . , n, χj = 0 ∀j > n.

We have:

a∗(i)wλ,χ = µiwλ,χ, i = 1, . . . , n, a∗(i)wλ,χ = 0, for i > n.

Proposition 8.1. Assume that λ 6= 0, µn 6= 0, λ = (λ, 0, 0, . . . ), µ = (µ1, µ2, · · · , µn) and

χ(z) =∑

i∈Z≤0

χiz−i−1 +

1

λ

n∑

i=1

µiz−i−1,

Then we have:

• M1(λ,µ) ∼= Lϕ(χ)⊗Πλ as modules for the Weyl vertex algebra M .• M1(λ,µ) has the structure of an irreducible Π(0)–module.

Remark 3. Note that if λ has not of the form (λ, 0, 0, . . . ), then the M–module structureon M1(λ,µ) can not be extended to a structure of Π(0)–module. We simply can not definethe vertex operator

YM1(λ,µ)(ec, z)(8.1)

in the usual sense. But we believe that the certain generalised vertex operator of type (8.1)can be constructed using irregular vertex operators as in the Gaiotto paper [11].

20 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

8.2. Free field realisation of L(d,λ,µ): the case M1(λ,µ) is a Π(0)–module. Westart with realisation of M1(λ,µ) as a Π(0)–module. Note that the operator δ(0) ∈End(M1(λ,µ)) commutes with the action of Mϕ(0)⊗Mγ,δ(1), and since M0 = W1+∞,−1 ⊂Mϕ(0)⊗Mγ,δ(1), we get that δ(0) commutes with the action of W1+∞,−1.

By construction we have that M1(λ,µ) is a module for the Heisenberg vertex algebraMγ,δ(1), generated by γ(z) and δ(z) = δ(z)−2ϕ(z), with the free action of δ(0) = δ(0)−2χ0.

For each d ∈ C, we get the following irreducible W1+∞,−1–module

L(d,λ,µ) =M1(λ,µ)

Jd,

where Jd = W1+∞,−1.(δ(0) − d)wλ,χ.

This implies that L(d,λ,µ) is an Mγ,δ(1)–module such that δ(0) acts as d · Id. Therefore

we have the following:

Proposition 8.2. Identify W1+∞,−1 as a subalgebra of the Heisenberg vertex algebra Mγ,δ(1).

Assume thatλ = (λ, 0, 0, . . . ), µ = (µ1, · · · , µn).

Then the Whittaker W1+∞,−1–module L(d,λ,µ) has the structure of an irreducible Whit-taker module for the Hiesenberg vertex algebra Mγ,δ(1), generated by the Whittaker vector

wλ,µ such that γ(0) ≡ −Id, δ(0) ≡ d · Id and

γ(i)wλ,µ = 0, δ(n)wλ,µ = −2µi

λwλ,µ (n ∈ Z>0).

8.3. The general case and non-tensor product modules. Proposition 8.2 shows thatif λ = (λ, 0, 0, . . . ), then L(d,λ,µ) has the structure of a module for the Heisenberg vertexalgebra M2(1). One can ask if L(d,λ,µ) in general are isomorphic to a Whittaker modulefor M2(1).

It is important note that M0 = W1+∞,c=−1∼= M(2) ⊗M1(1), and therefore Whittaker

modules for M0 can be obtained by using recent Tanabe paper [18] on Whittaker modulesfor the singlet algebra M(p), p ≥ 2. He proved in [18, Theorem 1.1] that every irreducibleM(p)–module which is generated by the Virasoro Whittaker vector can be realized as aWhittaker module for the Heisenberg vertex algebra M1(1).

Next result shows that in general L(d,λ,µ) is not realized as an irreducible, WhittakerM2(1)–module.

Proposition 8.3. Assume that λ = (λ0, . . . , λn), µ = (µ1, . . . , µm), n > 0 and λn, µm 6= 0.Then L(d,λ,µ) does not have the form (1.1), i.e, it is not realized as a Whittaker M2(1)–module.

Proof. For any v ∈ M1(λ,µ), denote by [v] the projection of v to L(d,λ,µ). Then [wλ,µ]

is the Whittaker vector L(d,λ,µ) for the Whittaker pair (gl, p) (cf. Subsection 2.4), whichis unique, up to a scalar factor.

Assume that L(d,λ,µ) ∼= Z1 ⊗ Z2 where Z1 is an irreducible Whittaker M(2)–module,and Z2 is an irreducible Whittaker MJ0(1)–module. An irreducible Whittaker MJ0(1)–module is generated by the Whittaker vector whheis such that for k ≥ 0:

J0(k)whheis = χkwhheis, χk ∈ C,

WHITTAKER MODULES FOR gl AND W1+∞ 21

and one easily sees that each vector in the Whittaker module must be locally finite forJ0(k), k ≥ 0. This implies that arbitrary vector w ∈ L(d,λ,µ) is locally finite for J0(k0)for all k0 > 0, meaning that

dim spanC{J0(k0)

iw| i ≥ 0} < ∞.(8.2)

We shall prove that it is not possible. We shall find k0 > 0 and w such that (8.2) does nothold. We take w = [wλ,µ].

Now we consider J0(k0)iwλ,µ, i ≥ 1. For simplicity we consider the case n ≥ m, when

k0 = n. We have

spanC{J0(k0)

i[wλ,µ]| i ≥ 0} = spanC{[a∗(0)iwλ,µ]| i ≥ 0}.

Assume that there are constants C0, . . . , Cm, Cm 6= 0 such that

C0[wλ,µ] + · · ·+ Cm[a∗(0)mwλ,µ] = 0.(8.3)

Set

u = C0wλ,µ + · · ·+ Cma∗(0)mwλ,µ ∈ M1(λ,µ).

Since1

λn(E0,m − λ0µm)mu = νwλ,µ (ν 6= 0),

we conclude that u is not in the ideal generated by (I − d)C[I], implying that [u] 6= 0. Thiscontradicts relation (8.3). Therefore spanC{J

0(k0)i[wλ,µ]| i ≥ 0} is infinite-dimensional,

which implies that L(d,λ,µ) does not have the form (1.1). The proof follows.�

Remark 4. In Appendix we shall present a different proof of Proposition 8.3 by proving thatL(d,λ,µ) is isomorphic an irreducible module for the Heisenberg-Virasoro algebra which isnot a tensor product module.

9. Generalized Whittaker modules for g = gl(2ℓ,C)

In this Section we take g = gl(2ℓ,C). Recall from Example 2.3 that g has triangulardecomposition g = n− ⊕ h⊕ n+. We take the subalgebra n ⊂ n+.

n = spanC{ei,j+ℓ |i, j = 1, . . . , ℓ}.

Now we are interested in Whittaker modules with respect to the pair (g, n), which we calledgeneralized Whittaker modules. In particular we are interested in a construction of simplequotients of the universal Whittaker module

Mλ = U(g)⊗U(n) Cvλ.

9.1. Weyl algebra A2ℓ and its Whittaker modules. Here we recall the definition ofthe Weyl algebra Aℓ. It is an complex associative algebra with generators

ai, a∗i , i = 1, . . . , 2ℓ

and relations

[ai, aj ] = [a∗i , a∗j ] = 0, [ai, a

∗j ] = δi,j , (i, j = 1, . . . , 2ℓ).

22 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

Define the normal ordering on A with

: xy :=1

2(xy + yx).

In paricular, we have

: aia∗i := aia

∗i −

1

2= a∗i ai +

1

2.

For α = (α1, . . . , αℓ), β = (β1, . . . , βℓ) ∈ Cℓ, we define the Whittaker module W (α, β) tobe the quotient

W (α,β) = A2ℓ/Iℓ(α,β),

where and Iℓ(α,β) is the left ideal

Iℓ(α,β) =⟨a1 − α1, . . . , aℓ − αℓ, a

∗ℓ+1 − β1, . . . , a

∗2ℓ − βℓ

⟩.

One shows that

Lemma 9.1. W (α,β) is an irreducible A2ℓ–module. It is generated by the Whittaker vectorwα,β = 1 + Iℓ(α,β) such that

aiwα,β = αiwα,β, a∗α+iwα,β = βiwα,β, i = 1, . . . , ℓ.

As a vector space

W (α,β) ∼= C[a∗1, · · · , a∗ℓ , aℓ+1, · · · , a2ℓ].

9.2. Embedding of g into A2ℓ. It is well known that there is a Lie algebra homomorphism

Φ : g → (A2ℓ)Lie

uniquely determined by

ei,j 7→: aia∗j : .

This implies that each A2ℓ–module can be treated as a g–module.

9.3. W (α,β) as a g–modules. Now we consider the generalized Whittaker modulesW (α,β)as g–modules.

Proposition 9.2. Assume that α 6= 0 and β 6= 0. W (α,β) is a Whittaker module for theWhittaker pair (g, n). In a particular, wα,β is a Whittaker vector such that

ei,ℓ+jwα,β = αiβjwα,β ∀i, j ∈ {1, . . . , ℓ}

and W (α,β) = U(g)wα,β.In particular, W (α,β) is a quotient of the universal Whittaker module Mλ with Whittaker

function:

λ = λα,β : n → C, λ(ei,j+ℓ) = αiβj , ∀i, j ∈ {1, . . . , ℓ}.

Proof. The proof is completely analogous to that presented in Proposition 5.1 below in the

case gl. �

WHITTAKER MODULES FOR gl AND W1+∞ 23

Note that the central element∑2ℓ

i=1 ei,i ∈ g acts on W (α,β) as

I =

2ℓ∑

i=1

: aia∗i : .

Since

Iwα,β =ℓ∑

i=1

(αia∗i + βiai+ℓ)wα,β,

which is not proportional to wα,β, we conclude that W (α,β) is reducible g–module.

Theorem 9.3. For each d ∈ C, α,β ∈ Cℓ, α,β 6= 0, we have

L(d,α,β) =W (α,β)

〈(I − d)C[I]wα,β〉

is an irreducible g–module.

Proof. Proof is analogous to that of Theorem 7.4. �

Appendix: M1(λ,µ) as a module for the Heisenberg-Virasoro algebra

In this section we prove a stronger statement that M1(λ,µ) is a cyclic module for theHeisenberg-Virasoro vertex subalgebra of W1+∞,−1. Our main argument will be that theaction of singlet M(2) on the Whittaker vector wλ,µ can be replaced by the action of thesinglet Virasoro subalgebra. Next we identify M1(λ,µ) and L(d,λ,µ) as restricted modulesfor the Heisenberg-Virasoro algebra recently classified in [20].

Recall that

Jk(z) = Y (a∗(−k)a, z) =∑

s∈Z

Jk(s)z−k−s−1.

The singlet algebra M(2) (cf. [2], [19]) is generated by the Virasoro vector L = J1 + 12 :

(J0)2 : +12DJ0 and by the primary field H of conformal weight 3. Recall that the Z2–

orbifold of M is isomorphic to the simple affine vertex algebra L−1/2(sl2) generated by

e = 12 : a2 :, h = −J0, f = −1

2 : (a∗)2 :,

the singlet algebra is isomorphic to the parafermion vertex algebra N−1/2(sl2) and H is a

scalar multiple of the parafermion generator W 3 in [9, Section 2]:

W 3 = k2h(−3)1 + 3kh(−2)h(−1)1 + 2h(−1)31− 6kh(−1)e(−1)f(−1)1

+3k2e(−2)f(−1)1 − 3k2f(−2)e(−1)1 (k = −1/2),

(see also [16, Section 4]).We have that LHV ir

c=−1 = LV irc=−2 ⊗M1(1), where LV ir

c=−2 denotes the simple Virasoro vertexalgebra of central charge c = −2 generated by L.

Theorem 9.4. We have:(1) M1(λ,µ) is an cyclic LHV ir

c=−1–module.

(2) L(d,λ,µ) is an irreducible LHV irc=−1–module.

24 DRAZEN ADAMOVIC, VERONIKA PEDIC TOMIC

(3) M1(λ,µ) and L(d,λ,µ) are not tensor product LHV irc=−1–modules. Moreover, there is a

nilpotent subalgebra P of H and 1–dimensional P–module U such that M1(λ,µ) = IndHPU .

Proof. The field H can be expressed using J0, J1 and it contains the non-trivial summandJ0(−1)31. Set H(i) := Hi+2, so we have

H(z) = Y (H, z) =∑

i∈Z

H(i)z−i−3.

By a direct calculation we get for s ∈ Z≥0:

H(3n + 3m+ s)wλ,µ = δs,0qwλ,µ

for certain q ∈ C, q 6= 0.Using the following relation in M(2)

3

4H(−6)1− L(−2)H(−4)1 +

3

2L(−3)H = 0,

and using the arguments as in [18, Lemma 3.1] we get that

M(2).wλ,µ = 〈L〉wλ,µ,

so M(2).wλ,µ is isomorphic to the Virasoro submodule obtained by the action of the Vira-soro generator L on the Whittaker vector wλ,µ. Since by Theorem 5.2 M1(λ,µ) is a cyclic

M(2) ⊗M1(1)–module, we conclude that M1(λ,µ) is a cyclic LHV irc=−1–module. This proves

assertion (1). The assertion (2) is a consequence of (1).Let us prove assertion (3). For simplicity we consider the case n + 1 ≥ m. Other cases

can be treat similarly.By a direct calculation we get for s ∈ Z>0:

J0(n+m+ s)wλ,µ = J1(n+m+ s)wλ,µ = 0,(9.1)

J0(n+m)wλ,µ = amwλ,µ, . . . , J0(n+ 1)wλ,µ = a1wλ,µ,(9.2)

J1(n+m)wλ,µ = bm+1wλ,µ, . . . , J1(n)wλ,µ = b1wλ,µ,(9.3)

where ai, bi ∈ C, am, bm+1 6= 0, and J0(n)wλ,µ, J1(n−1)wλ,µ are not proportional to wλ,µ.

Consider the nilpotent subalgebra P(n) of H spanned by J0(r + 1), J1(r), r ≥ n and

C1, C2. Then Cwλ,µ is the 1-dimensional P(n)–module satisfying relations (9.1)-(9.3) andC1wλ,µ = 2wλ,µ, C2wλ,µ = −wλ,µ. Then M1(λ,µ) and L(d,λ,µ) are quotients of theuniversal H–module:

IndHP(n)Cwλ,µ.

These universal modules and their simple quotients appeared in a slightly general form in[20, Section 7], where it was proved that they are not tensor product modules (see [20,Example 7.5, Remark A.4]). In particular, it was shown that for each d ∈ C:

IndHP(n)

U(H)(J0(0) − d)wλ,µ

is a simple, restricted H–module. This easily implies that as H–modules:

M1(λ,µ) = IndHP(n)Cwλ,µ, L(d,λ,µ) =

M1(λ,µ)

U(H)(J0(0) − d)wλ,µ.

WHITTAKER MODULES FOR gl AND W1+∞ 25

The proof follows. �

Corollary 9.5. W1+∞,c=−1–modules ρs(L(d,λ,µ)) are typical.

References

[1] D. Adamovic, Representations of the vertex algebra W1+∞ with a negative integer central charge,Comm. Algebra 29 (2001) no. 7, 3153–3166.

[2] D. Adamovic, Classification of irreducible modules of certain subalgebras of free boson vertex algebra,J. Algebra 270 (2003), 115–132.

[3] D. Adamovic, R. Lu, K. Zhao, Whittaker modules for the affine Lie algebra A(1)1 , Adv. Math. 289

(2016), 438–479.[4] D. Adamovic, C. H. Lam, V. Pedic, N. Yu, On irreducibility of modules of Whittaker type for cyclic

orbifold vertex algebras, J. Algebra 539 (2019), 1–23.[5] D. Adamovic, V. Pedic, On fusion rules and intertwining operators for the Weyl vertex algebra, J.

Math. Phys. 60 (2019), no. 8, 081701, 18 pp.[6] D. Adamovic, A. Milas, Logarithmic intertwining operators and W(2, 2p− 1)–algebras. J. Math. Phys.

48(7), 073503 (2007) 20 pp[7] P. Batra, V. Mazorchuk, Blocks and modules for Whittaker pairs, J. Pure Appl. Algebra 215, no. 7

(2011) 1552–1568.[8] T. Creutzig, A. Milas, False theta functions and the Verlinde formula, Adv. Math. 262, (2014) 520–545[9] C. Dong, C.H. Lam and H. Yamada, W-algebras related to parafermion algebras, J. Algebra 322 (2009)

2366–2403.[10] E. Frenkel, V. Kac, A. Radul, W. Wang, W1+∞ algebra and W(glN) with Central Charge N , Comm.

Math. Phys. 1995, 170, 337–357.[11] D. Gaiotto, Asymptotically free N=2 theories and irregular conformal blocks, arXiv:0908.0307[12] M. D. Gould and N. I. Stoilova: Casimir invariants and characteristic indentities for gl(∞), Journal of

Mathematical Physics 38, 4783 (1997); doi:10.1063/1.532123[13] V. Kac, A. Radul, Quasi–finite highest weight modules over the Lie algebra of differential operators on

the circle, Comm. Math. Phys. 157 (1993), 429–457[14] V. Kac, A. Radul, Representation theory of the vertex algebra W1+∞, Transform. Groups 1 (1996), no.

1-2, 41–70.[15] J. Lepowsky, H. Li, Introduction to vertex operator algebras and their representations, Progress in

Math. Vol. 227 Boston: Birkhauser, 2004.[16] D. Ridout, sl(2)−1/2 and the Triplet Model, Nucl. Phys.B835 (2010) 314-342[17] K. Tanabe, Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator

algebra of rank 1 by an automorphism of order 2 and Whittaker vectors, Proc. Amer. Math. Soc. 145(2017), no. 10, 4127–4140.

[18] K. Tanabe, Simple weak modules for some subalgebras of the Heisenberg vertex algebra and Whittakervectors, Algebr. Represent. Theory 23 (2020), no. 1, 53–66.

[19] W. Wang, W1+∞–algebra, W3–algebra, and Friedan–Martinec–Shenker bosonization Comm. Math.Phys., 195 (1998), pp. 95-111

[20] H. Tan, Y. Yao, K. Zhao, Simple restricted modules over the Heisenberg-Virasoro algebra as VOAmodules, arXiv:2110.05714v2 [math.RT]

D.A.: Department of Mathematics, Faculty of Science, University of Zagreb, Bijenicka 30, 10 000 Zagreb,Croatia; adamovic@math.hr

V.P.: Department of Mathematics, Faculty of Science, University of Zagreb, Bijenicka 30, 10 000 Zagreb,Croatia; vpedic@math.hr