Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks
Mikhail Bershtein, Roman Gonin
TL;DR
This paper constructs explicit bosonizations of certain quantum toroidal algebra representations, studies twisted $W$-algebras acting on them, and proves a conjectured relation on $q$-deformed conformal blocks related to $q$-deformed isomonodromy and CFT correspondence.
Contribution
It provides explicit bosonization formulas for Fock modules with nontrivial slope and establishes a key relation on $q$-deformed conformal blocks, advancing understanding of quantum toroidal algebras and $q$-deformed CFT.
Findings
Explicit bosonization of Fock modules with nontrivial slope
Analysis of twisted $W$-algebras acting on these modules
Proof of a conjectured relation on $q$-deformed conformal blocks
Abstract
We study certain representations of quantum toroidal algebra for . We construct explicit bosonization of the Fock modules with a nontrivial slope . As a vector space, it is naturally identified with the basic level 1 representation of affine . We also study twisted -algebras of acting on these Fock modules. As an application, we prove the relation on -deformed conformal blocks which was conjectured in the study of -deformation of isomonodromy/CFT correspondence.
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\FirstPageHeading
\ShortArticleName
Twisted Representations of Algebra of -Difference Operators
\ArticleName
Twisted Representations of Algebra
of -Difference Operators, Twisted - Algebras
and Conformal Blocks
\Author
Mikhail BERSHTEIN and Roman GONIN
\AuthorNameForHeading
M. Bershtein and R. Gonin
\Address
Landau Institute for Theoretical Physics, Chernogolovka, Russia \EmailDD[email protected]
\Address
Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia \EmailDD[email protected]
\Address
National Research University Higher School of Economics, Moscow, Russia
\Address
Institute for Information Transmission Problems, Moscow, Russia \Address Independent University of Moscow, Moscow, Russia
\ArticleDates
Received November 22, 2019, in final form August 01, 2020; Published online August 16, 2020
\Abstract
We study certain representations of quantum toroidal algebra for . We construct explicit bosonization of the Fock modules with a nontrivial slope . As a vector space, it is naturally identified with the basic level 1 representation of affine . We also study twisted -algebras of acting on these Fock modules. As an application, we prove the relation on -deformed conformal blocks which was conjectured in the study of -deformation of isomonodromy/CFT correspondence.
\Keywords
quantum algebras; toroidal algebras; -algebras; conformal blocks; Nekrasov partition function; Whittaker vector
\Classification
17B67; 17B69; 81R10
1 Introduction
Toroidal algebra. Representation theory of quantum toroidal algebras has been actively developed in recent years. This theory has numerous applications, including geometric representation theory and AGT relation [43], topological strings [1], integrable systems, knot theory [28], and combinatorics [13].
In this paper we consider only the quantum toroidal algebra; we denote it by U_{q,t}\big{(}\ddot{\mathfrak{gl}}_{1}\big{)}. The algebra depends on two parameters , and has PBW generators , and central generators , [12]. In the main part of the text we consider only the case , where toroidal algebra becomes the universal enveloping of the Lie algebra with these generators , , and the relation
[TABLE]
We denote this Lie algebra by , since there is a homomorphism from this algebra to the algebra of -difference operators generated by , with the relation ; namely .
There is another presentation of the algebra (and more generally U_{q,t}\big{(}\ddot{\mathfrak{gl}}_{1}\big{)}) using the Chevalley generators , , , see, e.g., [47].
In this paper we deal with the Fock representations of ; to be more precise there is a family of Fock modules, depending on the parameter (see Proposition 3.1 for a construction of ). They are just Fock representations of the Heisenberg algebra generated by . The images of and are vertex operators. A construction of this type is usually called bosonization.
It was shown in [18, 43] that the image of toroidal algebra U_{q,t}\big{(}\ddot{\mathfrak{gl}}_{1}\big{)} in the endomorphisms of the tensor product of Fock modules is the deformed -algebra for . There is the so-called conformal limit , in which deformed -algebras go to vertex algebras. These vertex algebras are tensor products of the Heisenberg algebra and the -algebras of . In the case , the central charge of the corresponding -algebra of is equal to . These -algebras appear in the study of isomonodromy/CFT correspondence (see [23, 24]). This is one of the motivations of our paper.
The -deformation of the isomonodromy/CFT correspondence was proposed in [9, 11, 31]. The main statement is an explicit formula for the -isomonodromic tau function as an infinite sum of conformal blocks for certain deformed -algebras with . In general, these tau functions are complicated, but there are special cases (corresponding to algebraic solutions) where these tau functions are very simple [5, 9]. These cases should correspond to special representations of -deformed -algebras. The construction of such representation is one of the purposes of this paper.
Twisted Fock modules. There is a natural action of on . We will parametrize by
[TABLE]
Then acts as
[TABLE]
For any module and , we denote by the module twisted by the automorphism (see Definition 2.5). The twisted Fock modules depend only on and (up to isomorphism). These numbers are the values of the central generators and , correspondingly, acting on . Therefore we will also use the notation for . Twisted Fock modules (for generic ) were used, for example, in [1] and [29].
In Section 4 we construct explicit bosonization of the twisted Fock modules for . Actually, we give three constructions: the first one in terms of -fermions (see Theorem 4.1), the second one in terms of -bosons (see Theorem 4.3) and the third one in terms of one twisted boson (see Theorem 4.4) (here, for simplicity, we assume that ). In other words, any twisted Fock module will be identified with the basic module for ; these two bosonizations correspond to homogeneous [21] and principal [32, 35] constructions.
The construction of the bosonization is nontrivial, because it is given in terms of Chevalley generators (note that the action is not easy to describe in terms of Chevalley generators). The appearance of affine is in agreement with the Gorsky–Neguţ conjecture [29]. More specifically, it was conjectured in [29] that there exists an action (with certain properties) of U_{p^{1/2}}\big{(}\widehat{\mathfrak{gl}}_{n}\big{)} on for ; we expect this to be -deformation of the -action constructed in this paper.
It is instructive to look at the formulas in the simplest examples. For simplicity, we give here only formulas for . Here we introduce the notation in a sloppy way (for details see Sections 3 and 4).
Example 1.1**.**
In the standard case , we have
[TABLE]
where , are complex conjugate fermions (see Section 3.2), is a boson and are generators of the Heisenberg algebra with relation (see Section 3.1).
Example 1.2**.**
The first nontrivial case is given by , . We have three formulas (corresponding to Theorems 4.1, 4.3 and 4.4):
[TABLE]
Here , and , are anticommuting pairs of complex conjugate fermions (see Section 4.1), are commuting bosons, and are generators of the Heisenberg algebra with the relation (see Section 4.2). The generators in (1.3) satisfy .
The relation between (1.1) and (1.2) is a standard boson-fermion correspondence. In the right-hand side of formula (1.3) we have only one Heisenberg algebra with generators , but since we have both integer and half-integer powers of , one can think that we have a boson with a nontrivial monodromy. This is the reason for the term ‘twisted boson’; we will also call this construction strange bosonization. Note that half-integer powers of cancel in the right-side of (1.3).
We present two different proofs of Theorems 4.1, 4.3 and 4.4. The first one is given in Section 5 and is based on the following idea. For any full rank sublattice of index , we have a subalgebra , which is spanned by for and central elements , . The algebra is isomorphic to ; the isomorphism depends on the choice of a positively oriented basis , in . Denote this isomorphism by .
If the basis , is such that , , then the restriction of the Fock module on is isomorphic to the sum of tensor products of the Fock modules
[TABLE]
where and . If we choose basis , in which differs from , by , we get an analogue of decomposition (1.4) with right-hand side given by a sum of tensor products of the twisted Fock modules. For the basis , , we write formulas for Chevalley generators of using either initial fermion or initial boson for . Applying this for the lattices with , we get Theorems 4.1, 4.3 and 4.4.
The secondproof of these theorems is based on the semi-infinite construction. Let denote the representation of the algebra in a vector space with basis for , where acts as -difference operators (see Definition 3.8). This representation is called vector (or evaluation) representation; the parameter is equal to . The Fock module is isomorphic to . After the twist, we get a semi-infinite construction of \mathcal{F}_{u}^{\sigma}\subset\big{(}\Lambda^{{}^{\!\infty/2}}\,V_{u}\big{)}^{\sigma}=\Lambda^{{}^{\!\infty/2}}\,(V_{u}^{\sigma}). Note that conjecturally the semi-infinite construction of can be generalized for (cf. [15]).
Twisted -algebras. Denote by the subalgebra of generated by and , for . There is an another set of generators of the completion of the U\big{(}\mathfrak{Diff}_{q}^{\geqslant 0}\big{)}, defined by the formula (see Appendix A for the definition of the power of ). The currents and for satisfy relations of the -deformed -algebra of (see [43]). We denote this algebra by .
There is an ideal in U\big{(}\mathfrak{Diff}_{q}^{\geqslant 0}\big{)}=\mathcal{W}_{q}(\mathfrak{gl}_{\infty}) which acts by zero on any tensor product , here . This ideal is generated by relations and
[TABLE]
where
[TABLE]
The quotient of is the -deformed -algebra of . We denote this algebra by ; it does no depend on (up to isomorphism) and acts on any tensor product (see [19, 43]).
In Section 7 we study a tensor product of the twisted Fock modules . We prove that the ideal generated by relations and
[TABLE]
acts by zero for .We denote the quotient by and call it the twisted -deformed -algebra of .
There exists another description of the above using the -deformed -algebra of introduced in [17]. Define by the formula
[TABLE]
The generators are elements of a localization of the completion of U\big{(}\mathfrak{Diff}_{q}^{\geqslant 0}\big{)}. These generators commute with and satisfy certain quadratic relations. The algebra generated by is denoted by .
There is an ideal in which acts by zero on any tensor product . This ideal contains relations , , and for . The quotient is a standard -algebra [17] (see also Definition 7.1). We have a relation , where is the Heisenberg algebra generated by .
In the case of a product of the twisted Fock modules the situation is similar. The corresponding ideal contains the relations , for . We present the quotient in terms of the generators and relations (this is Theorem 7.7). We call the algebra with such generators and relations by twisted -algebra ; see Definition 7.3.111One can find a definition of in [44, equations (37)–(38)]. The quadratic relations in the algebra are the same as in the untwisted case (see equation (7.1)–(7.2)), the only difference lies in the relation .
The algebra is graded, with . Let us rename the generators by T^{tw}_{k}[r]=T_{k}\big{[}r-\frac{n^{\prime}k}{n}\big{]}, for . The presentations of the algebra in terms of generators and the presentations of the algebra is terms of generators are given by the same formulas; the only difference is the region of . Heuristically, one can think that is the same algebra as but with currents having nontrivial monodromy around zero.
In order to explain these results in more details, consider an example of .
Example 1.3**.**
As a warm-up, consider the untwisted case . The algebra is -deformed Virasoro algebra [45]. It has one generating current and the relation reads
[TABLE]
where are coefficients of a series \sum\limits_{l=0}^{\infty}f[l]x^{l}=\sqrt{(1-qx)\big{(}1-q^{-1}x\big{)}}/(1-x). This algebra has a standard bosonization [45]
[TABLE]
where and are the generators of the Heisenberg algebra ; one can also add related to the parameter . In terms of the toroidal algebra this formula corresponds to the tensor product of two Fock modules , here .
Example 1.4**.**
Now, consider the twisted case . The algebra is generated by one current . The generators satisfy relation (1.5). The algebra is called twisted -deformed Virasoro algebra.
As was explained above, the representations of come from the twisted Fock modules The bosonization of the twisted Fock module leads to the bosonization of the . Using formula (1.2) we get a bosonization
[TABLE]
Using formula (1.3) we get a strange bosonization
[TABLE]
Here , and are modes of the odd Heisenberg algebra, . These formulas for bosonization are probably new.
Example 1.5**.**
One can also use embedding in order to construct a bosonization of the -algebras. Namely one can take a representation of with known bosonization and then express the -algebra related to in terms of these bosons.
For example, consider generated by , and the Fock representations of . One can show (for example, using (1.4)) that algebra related to acts on through the quotient . Therefore, we get an odd bosonization of non-twisted -deformed Virasoro algebra
[TABLE]
Here are the odd modes of the initial boson for . The even modes of the boson disappear in the formula since it belongs to .
It follows from the decomposition (1.4) that formula (1.7) gives bosonization of certain special representation , to be more specific, a direct sum of Fock modules (defined by (1.6)) with particular parameters for .
In the conformal limit formula (1.7) goes to the odd bosonization of the Virasoro algebra , see, e.g., [48].
Whittaker vectors and relations on conformal blocks. As an application, in Section 9 we prove the following identity
[TABLE]
Here the lattice is as above, (u;q,q)_{\infty}=\prod\limits_{i,j=0}^{\infty}\big{(}1-q^{i+j}u\big{)}. The function is a Whittaker limit of conformal block. By AGT relation it equals to the Nekrasov partition function. We recall the definition of below.
The relation (1.8) was conjectured in [5] in the framework of -isomonodromy/CFT correspondence. As we discussed in the first part of the introduction the main statement of this correspondence is an explicit formula for the -isomonodromic tau function as an infinite sum of conformal blocks. The left-hand side of (1.8) is a tau function corresponding to the algebraic solution of deautonomized discrete flow in Toda system, see [5, equation (3.11)]. The right-hand side of (1.8) is a specialization of conjectural formula [5, equation (3.6)] for the generic tau function of these flows. In differential case the isomonodromy/CFT correspondanse is proven in many cases, see [7, 25, 26, 30], but in the -difference case the main statements are still conjectures. The generic formula for tau function of deautonomized discrete flow in Toda system is proven only for particular case [10, 37]. Here we prove formula for arbitrary but for special solution.
Let us recall the definition of . The Whittaker vector is a vector in a completion of , which is an eigenvector of for with certain eigenvalues depending on , see Definition 9.2. Such vector exists and unique for generic values of . This property looks to be a part of folklore, we give a proof of this in Appendix D. The proof is essentially based on the results of [42, 43]. The function is proportional to a Shapovalov pairing of two Whittaker vectors
[TABLE]
We give a proof of (1.8) using decomposition (1.4). We consider the Whittaker vector for the algebra . Its Shapovalov pairing gives the left-hand side of the relation (1.8). On the other hand, we prove that its restriction to summands is the Whittaker vector for the algebra . So taking the Shapovalov pairing we get the right-hand side of the relation (1.8).
In the conformal limit the analogue of the relation (1.8) in case was proven in [8] by a similar method. The conformal limit of the decomposition (1.4) was studied in [4].
Discussion of case. As we mentioned above, is a specialization of quantum toroidal algebra U_{q,t}\big{(}\ddot{\mathfrak{gl}}_{1}\big{)} for . It is much more interesting to study the algebra without the constrain. Let us discuss our expectations on generalizations of the results from this paper.
It is likely that fermionic construction (see Theorem 4.1) will be generalized after the replacement of the fermions by vertex operators of quantum affine . Hence we have bosonization, expressing the currents in terms of exponents dressed by screenings. We also expect that representations of twisted and non-twisted -algebras can be realized via these vertex operators (see [6] for the case). It is not clear how one can generalize strange bosonization and connection with isomonodromy/CFT correspondence for .
Plan of the paper. The paper is organized as follows.
In Section 2 we recall basic definitions and properties on the algebra .
In Section 3 we recall basic constructions of the Fock module .
In Section 4 we present three constructions of the twisted Fock module : the fermionic construction in Theorem 4.1, the bosonic construction in Theorem 4.3, and the strange bosonic construction in Theorem 4.4.
In Section 5 we study restriction of the Fock module to a subalgebra . Using these restrictions we prove Theorems 4.1, 4.3 and 4.4.
In Section 6 we give an independent proof of Theorem 4.1 using the semi-infinite construction.
In Section 7 we study twisted -deformed -algebras. We define by generators and relations. Then we show in Theorem 7.7 that the tensor product is isomorphic to the certain quotient of ; we denote this quotient by . We show that acts on the tensor product of twisted Fock modules . At the end of the section we study relation between these modules and the Verma modules for and .
In Section 8 we prove decomposition (1.4). Then we study the strange bosonization of -algebra modules arising from the restriction of Fock module on .
In Section 9 we recall definitions and properties of Whittaker vector, Shapovalov pairing, and conformal blocks. Then we prove (1.8), see Theorem 9.30.
In Appendix A we give a definition and study necessary properties of regular product of currents for .
Appendices B and C consist of calculations which are used in Section 7.
In Appendix D we study the Whittaker vector for in the completion of the tensor product . We prove its existence and uniqueness (we use this in Section 9). To prove existence we present a construction of Whittaker vector via an intertwiner operator from [1]. We also relate this Whittaker vector to the Whittaker vector of introduced in [46].
2 -difference operators
In this section we introduce notation and recall basic facts about algebra , see [14, 27, 33].
Definition 2.1**.**
The associative algebra of -difference operators is an associative algebra generated by and with the relation .
Definition 2.2**.**
The algebra of -difference operators is a Lie algebra with a basis (where ), and . The elements and are central. All other commutators are given by
[TABLE]
Remark 2.3**.**
Note that the vector subspace of spanned by (for ) is closed under commutation, i.e., has a natural structure of Lie algebra (denote this Lie algebra by ). Consider a basis of this Lie algebra . Finally, is a central extension of by two-dimensional abelian Lie algebra spanned by and .
2.1 action
In this section we will define action on . Let be an element of corresponding to a matrix
[TABLE]
Then acts as follows
[TABLE]
Proposition 2.4**.**
Formula (2.2) defines action on by Lie algebra automorphisms.
Proof.
Note that (2.1) is covariant. ∎
For any -module denote by the corresponding homomorphism.
Definition 2.5**.**
For any -module and let us define the representation as follows. and are the same vector space with different actions, namely .
We will refer to as a twisted representation. More precisely, is the representation , twisted by .
2.2 Chevalley generators and relations
The Lie algebra is generated by , and . We will call them the Chevalley generators of . Define the following currents (i.e., formal power series with coefficients in )
[TABLE]
Let us also define the formal delta function
[TABLE]
Proposition 2.6**.**
Lie algebra is presented by the generators , for all , (for ), , and the following relations
[TABLE]
One can find a proof of Proposition 2.6 in [38, Theorem 2.1] or [47, Theorem 5.5].
3 Fock module
In this section we review basic constructions of representations of with and . These construction were studied in [27].
3.1 Free boson realization
Introduce the Heisenberg algebra generated by (for ) with relation . Consider the Fock module generated by such that for , .
Proposition 3.1**.**
The following formulas determine an action of on :
[TABLE]
We will denote this representation by .
Remark 3.2**.**
Note that does not appear in formulas (3.1)–(3.3). But we will need operator later (see the proof of Proposition 3.12) for the boson-fermion correspondence. Heuristically, one can think that .
Remark 3.3** (on our notation).**
In this paper, we consider several algebras and their action on the corresponding Fock modules. We choose the following notation. All these representations are denoted by the letter F (for Fock) with some superscript to mention an algebra. Since is the most important algebra in our paper, we use no superscript for its representation. Also, let us remark that we consider several copies of the Heisenberg algebra. To distinguish their Fock modules, we write a letter for generators as a superscript.
The standard bilinear form on is defined by the following conditions: operator is dual of , the pairing of with itself equals 1. We will use the bra-ket notation for this scalar product. For an operator we denote by the scalar product of with .
Proposition 3.4**.**
Suppose the algebra acts on so that and ; . Then this representation is isomorphic to .
Proof.
Consider the current
[TABLE]
It is easy to verify that . Since is irreducible, for some formal power series with -coefficients. On the other hand, . This implies (3.2). The proof of (3.3) is analogous. ∎
Proposition 3.5**.**
Denote . The action of on Fock representation is given by the following formula
[TABLE]
Proof.
The commutation relation (2.1) implies that formula (3.4) holds up to a pre-exponential factor. Also, we see from (2.1) that
[TABLE]
The factor can be found inductively from (3.5). ∎
3.2 Free fermion realization
In this section we give another construction for the Fock representation of . To do this, let us consider the Clifford algebra, generated by and for subject to the relations
[TABLE]
Consider the currents
[TABLE]
Consider a module with a cyclic vector and relation
[TABLE]
The module is independent of . The isomorphism can be seen from the formulas and . Let us define the -dependent normal ordered product (to be compatible with ) by the following formulas
[TABLE]
Proposition 3.6**.**
The following formulas determine an action of on :
[TABLE]
Let us denote this representation by .
Remark 3.7**.**
The Products \psi\big{(}q^{-1/2}z\big{)}\psi^{*}\big{(}q^{1/2}z\big{)} and \psi\big{(}q^{1/2}z\big{)}\psi^{*}\big{(}q^{-1/2}z\big{)} from formulas (3.9)–(3.10) are not normally ordered (see Appendix A for a formal definition and some other technical details on the regular product). In particular, this reformulation implies that does not depend on .
3.3 Semi-infinite construction
Definition 3.8**.**
The evaluation representation of the algebra is a vector space with the basis for and the action
[TABLE]
Remark 3.9**.**
The associative algebra acts on . The representation of is obtained via evaluation homomorphism .
Remark 3.10**.**
Informally, one can consider as for . Define the action of as follows. The generator acts by multiplication and . However, is not well defined for arbitrary complex . So we consider as a parameter of representation instead of .
Let us consider the semi-infinite exterior power of the evaluation representation . It is spanned by , where is a Young diagram and . Let and be Frobenius coordinates of .
Proposition 3.11**.**
There is a -modules isomorphism \Lambda^{{}^{\!\infty/2}}\,V_{u}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{\scriptstyle\sim}}\,}\mathcal{M}_{u} given by
[TABLE]
Proposition 3.12**.**
There is an isomorphism of -modules . The submodule is spanned by .
Proof.
Recall the ordinary boson-fermion correspondence (see [34]). The coefficients of
[TABLE]
are indeed generators of the Heisenberg algebra. Moreover, . The highest vector of is (in particular, ). Note that this is the decomposition of -modules as well. Also, note that
[TABLE]
Therefore one can use Proposition 3.4 for each summand . ∎
There is a basis in the Fock module given by semi-infinite monomials
[TABLE]
To write the action of in this basis, let us remind the standard notation. Let be the number of non-zero rows. We will write for the th box in the th row (i.e., ). The content of a box . For the diagram , we define a skew Young diagram , being a set of boxes in which are not in . Ribbon is a skew Young diagram without squares. The height of a ribbon is one less than the number of its rows.
Proposition 3.13**.**
The action of on is given by the following formulas
[TABLE]
here .
In particular,
[TABLE]
Let us introduce the notation . Define an operator by the following formula
[TABLE]
The operator was introduced in [3] and is well known nowadays.
Proposition 3.14**.**
The operator enjoys the property .
Proof.
Corollary 3.15**.**
* for .*
Remark 3.16**.**
Also, Corollary 3.15 follows from Proposition 3.4: we will use this approach to prove Proposition 5.2.
Corollary 3.17**.**
The twisted representation is determined up to isomorphism by and .
Proof.
Corollary 3.15 implies that . Note that
[TABLE]
For the fixed and , all the possible choices of and appear for the appropriate . ∎
4 Explicit formulas for twisted representation
In this section we provide three explicit constructions of twisted Fock module for
[TABLE]
Constructions are called fermionic, bosonic, and strange bosonic. This section contains no proofs. We will give proofs in Sections 5. In Section 6 we will provide an independent proof of Theorem 4.1.
4.1 Fermionic construction
We need to consider the -graded th tensor power of the Clifford algebra defined above. More precisely, consider an algebra generated by and , for ; , subject to relations
[TABLE]
Consider the currents
[TABLE]
Consider a module with a cyclic vector and the relations
[TABLE]
The module does not depend on . The isomorphism can be seen from the following formulas:
[TABLE]
Theorem 4.1**.**
The formulas below determine an action of on
[TABLE]
The module obtained is isomorphic to .
Since , we have obtained a fermionic construction for .
4.2 Bosonic construction
Let us consider the th tensor power of the Heisenberg algebra. More precisely, this algebra is generated by for and with the relation . Let us extend the algebra by adding the operators , obeying the following commutation relations. The operator commutes with all the generators except for and satisfy . Denote
[TABLE]
Remark 4.2**.**
Informally, one can think that there exists an operator satisfying . However, this operator will not act on our representation. We will use as a formal symbol. Our final answer will consist only of , but not of without the exponent.
We need a notion of a normally ordered exponent . The argument of a normally ordered exponent is a linear combination of and . Let , , , and denote a linear combination of for , for , and , correspondingly ( is not fixed). Define
[TABLE]
Also, note that will have the coefficient . We shall understand it formally; the action of the operator is well defined, since in the representation to be considered below, acts as multiplication by an integer at each Fock module.
Let be a lattice with the basis . Consider the group algebra \mathbb{C}\big{[}\mathbf{Q}_{(n)}\big{]}. This algebra is spanned by for . Let us define the action of the commutative algebra generated by on \mathbb{C}\big{[}\mathbf{Q}_{(n)}\big{]}:
[TABLE]
Let be the Fock representation of the algebra generated by for ; i.e., there is a cyclic vector such that for .
Finally, we can consider F^{na}\otimes\mathbb{C}\big{[}\mathbf{Q}_{(n)}\big{]} as representation of the whole Heisenberg algebra as follows: for acts on the first factor, acts on the second factor. Also, \mathbb{C}\big{[}\mathbf{Q}_{(n)}\big{]} acts on F^{na}\otimes\mathbb{C}\big{[}\mathbf{Q}_{(n)}\big{]}.
Theorem 4.3**.**
There is an action of on F^{na}\otimes\mathbb{C}\big{[}\mathbf{Q}_{(n)}\big{]} determined by the formulas
[TABLE]
here we consider the product over such that for and for .
The representation obtained is isomorphic to .
4.3 Strange bosonic construction
We will use notation of Section 3.1. Let be a th primitive root of unity, e.g., .
Theorem 4.4**.**
There is an action of on determined by the formulas
[TABLE]
The representation obtained is isomorphic to .
As before the representation does not depend on , see Remark 3.2.
5 Twisted representation via a sublattice
5.1 Sublattices and subalgebras
Consider a full rank sublattice of index (i.e., is a finite group of order ). Let us define a Lie subalgebra which is spanned by for and central elements , .
Denote by , , standard generators of . Let and be a basis of . Define a map
[TABLE]
Proposition 5.1**.**
Let , be a positively oriented basis i.e., . Then the map is a Lie algebra isomorphism .
Proof.
It follows from (2.1) directly. ∎
Slightly abusing notation, denote the Fock representation of by .
Proposition 5.2**.**
Let and . Then as -modules.
Proof.
Note that the Fock module is -graded with grading given by
[TABLE]
Recall that a character of a -graded module is the generating function of dimensions of the graded components. Then the character of Fock module \operatorname{ch}\mathcal{F}_{u}=1/(\mathfrak{q})_{\infty}:=\prod\limits_{k=1}^{\infty}1/\big{(}1-\mathfrak{q}^{k}\big{)}.
Consider a subalgebra in spanned by and . Note that is isomorphic to the Heisenberg algebra. Since , the character of the -Fock module is also ; i.e., it coincides with . This implies that restricted to is isomorphic to the -Fock module. To finish the proof, we use Propositions 3.4 and 3.5. ∎
5.2 Twisted Fock vs restricted Fock
From now on we change . Our goal is to construct an action of on the Fock module twisted by as in (4.1) for . Consider a sublattice spanned by and . Consider another basis of obtained by
[TABLE]
Remark 5.3**.**
The construction of the sublattice naturally appears, if one require to be a transition matrix from to and assume , .
Denote the Fock module of by .
Theorem 5.4**.**
There is an isomorphism of -modules .
Proof.
Proposition 5.2 implies . On the other hand, relations (5.1) and (5.2) yield that is the transition matrix from , to , . ∎
Corollary 5.5**.**
There is an isomorphism of -modules .
Theorem 5.4 combined with results from Section 3 enables us to find explicit formulas for action on . We will do this below.
5.3 Explicit formulas for restricted Fock
5.3.1 Fermionic construction via sublattice
Denote fermionic representation of by . To be more specific, let us rewrite formulas from Section 3.2 for .
[TABLE]
Proposition 5.6**.**
The following formulas below determine an action of on
[TABLE]
The module obtained is isomorphic to for , .
Remark 5.7**.**
Below we will substitute to prove Theorem 4.1. However, Proposition 5.6 is more general, than it is necessary for the proof, since we do not assume here that . We will need the case of arbitrary in Section 8.
Proof.
We use the notation , and for the Chevalley generators of . The generators of (identified by ) will be denoted by , and . Let us write the identification explicitly for the Chevalley generators
[TABLE]
Let us consider currents and for . These currents are defined by following equality
[TABLE]
Let us denote the modes of and as in equality (4.1). It is easy to see that these modes satisfy Clifford algebra relations (4.2), (4.3). So we have identified the Clifford algebra and the th power of Clifford algebra. This leads to an identification .
Substituting (5.11) into (5.3) and (5.4), we obtain
[TABLE]
For technical reasons, we need to treat the cases and separately. Let us first consider the case . Using formulas (5.8)–(5.10), we see that
[TABLE]
For we obtain
[TABLE]
This can be rewritten as
[TABLE]
Note that the -dependent normal ordering is defined in terms of and . One can check (cf. (A.3))
[TABLE]
Hence
[TABLE]
Proof of Theorem 4.1.
Follows from Theorem 5.4 and Proposition 5.6. ∎
5.3.2 Bosonic construction via sublattices
Proposition 5.8**.**
There is an action of on determined by the following formulas
[TABLE]
here we consider the product over such that for and for .
The representation obtained is isomorphic to for , .
Proof Proposition 5.8.
We need an upgraded version boson-fermion correspondence for the proof. Namely, there is an action of th tensor power of the Heisenberg algebra on given by
[TABLE]
Let be a lattice spanned . According to boson-fermion correspondence .
Lemma 5.9**.**
Vector subspace is a -submodule with respect to action, defined in Proposition 5.6). The action of on the subrepresentation is given by (5.13)–(5.15).
Proof.
One should substitute
[TABLE]
into fermionic formulas (5.5)–(5.7). ∎
Recall that decomposition of is given by eigenvalues of ; more precisely, operator acts by on .
Lemma 5.10**.**
Using identification cf. (5.11), we obtain .
Proof.
This follows from \big{\lceil}\frac{l-b}{n}\big{\rceil}=0 for and (cf. (5.12)). ∎
Lemma 5.10 implies that the identification of vector spaces leads to identification of subspaces . Let us package identifications of vector subspaces into a commutative diagram
[TABLE]
Proposition 5.6 states that formulas (5.5)–(5.7) gives an action of with respect to identification of bottom line of the diagram. Therefore, Lemma 5.9 implies that formulas (5.13)–(5.15) describes the action of with respect to identification of top line of the diagram. ∎
Proof of Theorem 4.3.
Follows from Theorem 5.4 and Proposition 5.8. ∎
5.3.3 Strange bosonic construction via sublattices
Proposition 5.11**.**
There is an action of on defined by formulas
[TABLE]
Obtained module is isomorphic to for , .
Proof.
[TABLE]
Substitution of -version of (3.1)–(3.3) to (5.18)–(5.20) finishes the proof. ∎
Proof of Theorem 4.4.
Follows from Theorem 5.4 and Proposition 5.11. ∎
6 Twisted representation via a Semi-infinite construction
This section is devoted to another proof of the Theorem 4.1. So we use the same notation
[TABLE]
Twisted evaluation representation. Let be a matrix unit (all entries are 0 except for one cell, where it is 1; this cell is in th column and th row).
Consider a homomorphism defined by
[TABLE]
Algebra tautologically acts on \mathbb{C}^{n}\big{[}z,z^{-1}\big{]}. Therefore, homomorphism induces an action of on .
Proposition 6.1**.**
Obtained representation of is isomorphic to .
Proof.
Consider a basis of evaluation representation \mathbb{C}\big{[}x,x^{-1}\big{]}^{\sigma}. Action with respect to this basis looks like
[TABLE]
Let be such numbers that and . Substituting into (6.3) we obtain
[TABLE]
Let us identify \mathbb{C}^{n}\big{[}z,z^{-1}\big{]}\xrightarrow{\sim}\mathbb{C}\big{[}x,x^{-1}\big{]} by . Then formula (6.4) will be rewritten
[TABLE]
To be compared with formula (6.1) this proves the proposition for . The proof for is analogous. For proposition is obvious from (6.2). ∎
Semi-infinite construction. To apply semi-infinite construction we need to pass from associative algebras to Lie algebras.
Definition 6.2**.**
Algebra is a Lie algebra with basis (where and ), and . Elements and are central. All other commutators are given by
[TABLE]
Proposition 6.3**.**
There is an action of on given by formulas
[TABLE]
Obtained representation is isomorphic to \Lambda^{{}^{\!\infty/2}}\,\mathbb{C}^{n}\big{[}z,z^{-1}\big{]}.
Proof of Theorem 4.1.
According to Proposition 3.11, . Then . Therefore, Propositions 6.1 and 6.3 imply Theorem 4.1. ∎
7 --algebras
7.1 Definitions
Topological algebras and completions. In this section we will work with topological algebras. Let us define topological algebra appearing as a completion of . It is given by projective limit of where is the left ideal generated by non-commutative polynomials in of degree (with respect to grading ). Although each does not have a structure of algebra, so does the projective limit. Moreover, the projective limit has natural topology.
Below we will ignore all corresponding technical problems concerning completions and topology. We will use term ‘generators’ instead of ‘topological generators’, the same notation for , and its completion and so on.
7.1.1 Non-twisted -algebras
Let us introduce a notation
[TABLE]
Definition 7.1**.**
Algebra is generated by for and . It is convenient to add generators . The defining relations are
[TABLE]
Introduce currents . Then relations (7.1)–(7.2) can be rewritten in current form
[TABLE]
Also note that .
Remark 7.2**.**
There are different approaches to definition of --algebra. For example, in [17] algebra was defined via bosonization. The currents satisfy relation [17, Theorem 2]
[TABLE]
where
[TABLE]
One can check that limit gives relation (7.3). However [17] do not provide presentation of in terms of generators and relations.
In the paper [43] relation (2.62) defines algebra which (non-essentially) differs from mentioned above (and from defined above).
7.1.2 Twisted --algebras
Twisted --algebra depends on remainder of modulo . If , then we get definition of non-twisted --algebra from last section. One can find definition of in [44, equations (37)–(38)].
Definition 7.3**.**
Algebra is generated by for and . It is convenient to add . The defining relations are
[TABLE]
Let us rewrite relations (7.5)–(7.6) in the current form. Define currents
[TABLE]
Note that
[TABLE]
Proposition 7.4**.**
Relation (7.5) is equivalent to
[TABLE]
Relation (7.6) is equivalent to
[TABLE]
Remark 7.5**.**
In non-twisted case we have relations (7.3) and (7.4) for currents . In twisted case we have the same relations, but for two different sets of currents and . One should also keep in mind (7.7).
7.2 Connection of with
Connection between and is known (see [19, Proposition 2.14] or [43, Proposition 2.25]). In this section we generalize it for arbitrary .
Let be a Heisenberg algebra generated by with relation \big{[}\tilde{H}_{i},\tilde{H}_{j}\big{]}=ni\delta_{i+j,0}. We will prove that there is a surjective homomorphism . Secretly, generators are mapped to under the homomorphism. Let us introduce a notation to describe this homomorphism more precisely.
Define
[TABLE]
Also, let introduce notation
[TABLE]
Define
[TABLE]
Note that commute with .
Let be two sided ideal in generated by , and (here ). Parameter is not essential since automorphism maps to . So we will abbreviate .
Lemma 7.6**.**
* for .*
Proof.
It holds in
[TABLE]
On the other hand,
[TABLE]
Hence, . ∎
Theorem 7.7**.**
There is an algebra isomorphisms such that
[TABLE]
The map in opposite direction is given by
[TABLE]
The rest of this section is devoted to proof of Theorem 7.7. First of all, we will prove that formula (7.11) indeed defines a homomorphism (see Proposition 7.12). Then we prove that formulas (7.12)–(7.14) defines a homomorphism in opposite direction (see Proposition 7.13). Finally, we note that maps and are mutually inverse.
Proposition 7.8**.**
Currents as power series with coefficients in satisfy relation (7.8).
Proof.
Let us define power series in two variables
[TABLE]
According Corollary B.6, is regular in sense of Definition A.2. Following relations follows from results of Appendix A
[TABLE]
More precisely, (7.15)–(7.16) easily follows from Propositions A.8. One can find a proof of (7.17) at the end of Appendix A.
It is straightforward to check that
[TABLE]
Formulas (7.18) and (7.15)–(7.17) implies that
[TABLE]
Lemma 7.9**.**
The following OPE holds in
[TABLE]
or, equivalently
[TABLE]
Proof.
Denote by . We will write ; this definition reminds standard Definition A.3, but is applied in different situation ( is not a current in one variable). Note that
[TABLE]
Recall that . It follows from (2.4)that
[TABLE]
Using identity
[TABLE]
we obtain
[TABLE]
Finally, we conclude that
[TABLE]
Relation (7.19) follows from (7.21) and (7.22). ∎
Proposition 7.10**.**
In algebra holds
[TABLE]
Proof.
Using relation , we find a relation in
[TABLE]
Comparing coefficient of with (7.19), we obtain
[TABLE]
Denote by .
Proposition 7.11**.**
Following relation holds in
[TABLE]
Proof.
Proposition 7.10 imply
[TABLE]
It is straightforward to finish the proof using (7.20). ∎
Proposition 7.12**.**
Formula (7.11) defines a homomorphism from to the algebra .
Proof.
Evidently, and commute, and form a Heisenberg algebra. We only have to check that form algebra. Relation of algebra follows from Propositions 7.8 and 7.11. ∎
Proposition 7.13**.**
Formulas (7.12)–(7.14) defines a homomorphism from to the algebra .
Proof.
Let us check that these formulas define morphism from . According to Proposition 2.6, it is enough to prove relations (2.3)–(2.8). It is done in Appendix C. Evidently, annihilates . ∎
Proposition 7.14**.**
Maps and are mutually inverse.
Proof.
Let us prove first. The algebra is generated by modes of . Hence it is sufficient to check and \mathcal{P}\mathcal{S}\big{(}T_{1}(z)\big{)}=T_{1}(z). Both of them are straightforward.
The algebra is generated by modes of and . Evidently, \mathcal{S}\mathcal{P}\big{(}E(z)\big{)}\allowbreak=E(z). Proposition 7.10 implies \mathcal{S}\mathcal{P}\big{(}F(z)\big{)}=F(z). ∎
Proof of Theorem 7.7.
Follows from Propositions 7.12, 7.13 and 7.14 ∎
7.3 Bosonization of
Let be as in (4.1). Corollary 3.17 states that representation actually does not depend on and ; it is determined by and . Let us denote the representation by .
7.3.1 Fock representation via
In this section we will discuss connection of twisted - algebras and twisted representations .
Lemma 7.15**.**
In representation operator acts by .
Proof.
We will use formula (4.4) to calculate .
By Proposition A.8
[TABLE]
for any , (even for ).
Consider a sequence of numbers such that . Thus,
[TABLE]
Using bosonization (5.16), (5.17) we obtain (cf. (7.9) and (7.10))
[TABLE]
Consequently,
[TABLE]
Proposition 7.16**.**
Suppose, are representation of such that ideal acts by zero for . Then acts by zero on for , and .
Proof.
Recall that by Lemma 7.6. Thus, act on as
[TABLE]
Proposition 7.17**.**
Ideal acts by zero on for
[TABLE]
Proof.
Follows from Lemma 7.15 and Proposition 7.16. ∎
Theorem 7.18**.**
There is an action of on such that action of factors through .
Proof.
According to Proposition 7.17, algebra acts on . By Theorem 7.7, algebra is isomorphic to . ∎
Remark 7.19**.**
One can consider tensor product of Fock modules with different twists . According to Proposition 7.16 algebra \mathcal{W}\big{(}\mathfrak{sl}_{\sum n_{i}},\sum n^{\prime}_{i}\big{)}\otimes U(\mathfrak{Heis}) acts on this space. Obtained representation is ‘irregular’ (cf. [39]). In Section D.2 we consider an intertwiner between irregular and (graded completion of) regular representation.
7.3.2 Explicit formula for bosonization
Below we will write explicit formula for bosonization of . This bosonization comes from action of on . Recall that realization of is written via for .Denote by (for and ) generators of Heisenberg algebra action on th factor of the tensor product .
To write action of we need to introduce slightly different version of Heisenberg algebra. Namely, consider an algebra, generated by for ; and . Relations are given by linear dependence and commutation relations
[TABLE]
Let us define representation F^{\eta}\otimes\mathbb{C}\big{[}\mathbf{Q}^{d}_{(n)}\big{]} (cf. Section 4.2). Lattice consist of elements such that and for any it holds . Define an action
[TABLE]
First factor is a Fock space for subalgebra for . We can consider F^{\eta}\otimes\mathbb{C}\big{[}\mathbf{Q}^{d}_{(n)}\big{]} as representation of whole Heisenberg algebra as follows: for acts on first factor, acts on the second factor by (7.24). Note, that also \mathbb{C}\big{[}\mathbf{Q}^{d}_{(n)}\big{]} acts on F^{\eta}\otimes\mathbb{C}\big{[}\mathbf{Q}^{d}_{(n)}\big{]}. Let us introduce notation
[TABLE]
Proposition 7.20**.**
There is an action of on F^{\eta}\otimes\mathbb{C}\big{[}\mathbf{Q}^{d}_{(n)}\big{]} given by formulas
[TABLE]
here we consider product over such that for and for .
Proof.
Denote by \big{[}\mathcal{F}^{(n^{\prime},n)}_{u_{1}}\otimes\cdots\otimes\mathcal{F}_{u_{d}}^{(n^{\prime},n)}\big{]}_{h} subspace of such that for . According to Theorem 7.18, the algebra acts on \big{[}\mathcal{F}^{(n^{\prime},n)}_{u_{1}}\otimes\cdots\allowbreak\otimes\mathcal{F}_{u_{d}}^{(n^{\prime},n)}\big{]}_{h}.
On the other hand map is a homomorphism. Therefore, one can identify F^{\eta}\otimes\mathbb{C}\big{[}\mathbf{Q}^{d}_{(n)}\big{]} and \big{[}\mathcal{F}^{(n^{\prime},n)}_{u_{1}}\otimes\cdots\otimes\mathcal{F}_{u_{d}}^{(n^{\prime},n)}\big{]}_{h}.
Substitution of (4.5), (4.6), and (7.23) to (7.13), (7.14) finishes the proof. ∎
Denote obtained representation by .
Remark 7.21**.**
The parameter is determined by only up to -th root of unity. The modules with different are non-isomorphic in general (so notation is ambiguous). For example, one can see this from the highest weights defined in the next section.
The modules with different are related by an external automorphism of .
Example 7.22**.**
Let us consider case of twisted Virasoro algebra, i.e., , , . Then everything is expressed via one boson with relation and . So there is an action of on given by
[TABLE]
We can simplify the formula using conjugation by
[TABLE]
7.3.3 Explicit formulas for strange bosonization
To write formulas for strange bosonization we need to consider Heisenberg algebra generated by for and . Relations are given by linear dependence and commutation relations
[TABLE]
Denote corresponding Fock module by .
Proposition 7.23**.**
There is an action of on given by
[TABLE]
Obtained representation is isomorphic to .
Proof.
The proof is analogous to proof of Proposition 7.20. The only difference is that we have to use (4.7), (4.8) instead of (4.5), (4.6). This representation is isomorphic to since it also corresponds to . ∎
7.4 Verma modules vs Fock modules
Connection of Fock module and Verma module is known in non-twisted case. In this Subsection we will generalize it for . Denote .
Definition 7.24**.**
.
Denote modes of by . Namely, .
Definition 7.25**.**
Verma module is a module over with cyclic vector and relations
[TABLE]
Consider a grading on given by . Verma module is a graded module with grading defined by requirement .
Proposition 7.26**.**
* is spanned by for and .*
Proof.
Recall that . Therefore acts on .
Lemma 7.27**.**
Module is spanned by .
Sketch of the proof.
Let us consider operator defined by . On one hand is a basis of (this basis coincide with a basis of complete homogeneous polynomials up to renormalization of Heisenberg algebra generators). One the other hand,
[TABLE]
here lower terms are taken with respect to lexicographical order. ∎
Remark 7.28**.**
Lemma 7.27 holds for any exponent :\!\exp\big{(}\sum\alpha_{j}H_{j}z^{-j}\big{)}\!: such that for . The proof does not use any other properties of coefficients . One can find the proof as the last part of proof of [43, Proposition 2.29].
Define and as subalgebras of spanned by with and correspondingly.
Lemma 7.29**.**
Vector is cyclic with respect to action of .
Proof.
Theorem 7.7 implies that the natural map U\big{(}\mathfrak{Diff}_{q}^{\ \geqslant 0}\big{)}\rightarrow U(\mathfrak{Diff}_{q})/J_{n,n_{tw}} is surjective. Hence Verma module is generated by non-commutative monomials in and applied to . Using relation [H_{j},E_{i}]=\big{(}q^{-j/2}-q^{j/2}\big{)}E_{i+j}, we see that the module is spaned by . Lemma 7.27 implies that the module is spanned by ∎
In [42, (3.48)] author states, that algebra has a PBW-like basis for . Thus is spanned by (with the same condition on and ).
Note that if then such vector is 0; moreover if then acts by multiplication on a constant, hence it can be excluded.Also note that for , therefore we assume . The Proposition 7.26 is proven. ∎
Corollary 7.30**.**
Coefficients of are less or equal to coefficients of \prod\limits_{k=1}^{\infty}\big{(}1-\mathfrak{q}^{\frac{kd}{n}}\big{)}^{-d}.
Theorem 7.31**.**
If is irreducible then natural map is an isomorphism for
[TABLE]
Proof.
Let us first prove existence of the map . This will follow automatically from the following lemma.
Lemma 7.32**.**
The highest vector satisfy following conditions
[TABLE]
Proof.
Assertions (7.26)–(7.27) are evident. Let us check (7.28). Denote by action of on th tensor multiple; in particular . Recall that . Thus
[TABLE]
here dots denote summands with a power which is not a multiple of (thus, this summands do not contribute to ). To finish the proof we note that
[TABLE]
Image of is whole , since is irreducible. Note that \operatorname{ch}\big{(}\mathcal{F}^{(n_{tw}/d,n/d)}_{u_{1}}\otimes\dots\otimes\mathcal{F}^{(n_{tw}/d,n/d)}_{u_{d}}\big{)}=\prod\limits_{k=1}^{\infty}\frac{1}{\big{(}1-q^{\frac{kd}{n}}\big{)}^{d}}. Corollary 7.30 implies that is injective and \operatorname{ch}\mathcal{V}^{\mathcal{W}_{q}(\mathfrak{gl}_{n},n_{tw})}_{\lambda_{1},\dots,\lambda_{d}}=\prod\limits_{k=1}^{\infty}\frac{1}{\big{(}1-q^{\frac{kd}{n}}\big{)}^{d}}. ∎
Remark 7.33**.**
Note that representation is irreducible iff is irreducible. In particular, for then is irreducible automatically. Generally, criterion of irreducibility of is given by Lemma 9.1.
Definition 7.34**.**
Verma module is a module over with cyclic vector and relations
[TABLE]
Introduce grading on by . Verma module is a graded module with grading defined by .
To simplify notation for comparison of and cases, we will assume below that (cf. Remark 7.21).
Proposition 7.35**.**
* with respect to identification .*
Proof.
The existence of maps in both directions can be checked directly using universal property of the Verma module. Evidently, these maps are mutually inverse. ∎
Corollary 7.36**.**
If is irreducible then natural map is an isomorphism for as in (7.25).
8 Restriction on for general sublattice
We generalize results of Section 5 for arbitrary sublattice. For applications in Section 9 we will need only case of sublattice .
8.1 Decomposition of restriction
Let be a sublattice of finite index. Let us choose basis , of lattice so that and . Let be the greatest common divisor of and .
Theorem 8.1**.**
There is an isomorphism of -modules
[TABLE]
Proof.
Proposition 3.5 implies that . Hence it is enough to consider case . We will use realization of constructed in Proposition 5.8. Strategy of our proof is as follows; first we will construct decomposition on the level of vector spaces and then study action on each direct summand.
For each let be a subset of lattice consisting of elements
[TABLE]
Note that
[TABLE]
Or equivalently
[TABLE]
Let be Fock module for Heisenberg algebra generated by for . Then
[TABLE]
Let us show that (8.2) is a decomposition of -modules (moreover, that it leads to decomposition (8.1)).Let us define
[TABLE]
here (product over such that for and for ).
Lemma 8.2**.**
Formulas (8.3)–(8.5) defines an action of on F^{\frac{n}{d}a,\alpha}\otimes\mathbb{C}\big{[}\mathbf{Q}_{(n/d)}^{\alpha,l_{\alpha}}\big{]}; obtained representation is isomorphic to .
Sketch of the proof..
Let us define (product over satisfying above inequalities and condition . One can check that there exists an index set such that conjugation of and by will turn to . Theorem 4.3 finishes the proof. ∎
On the other hand, formulas (5.13)–(5.15) implies
[TABLE]
Therefore embedding of vector space from first row of following commutative diagram leads to second row embedding of -modules
[TABLE]
Corollary 8.3**.**
Following -modules are isomorphic
[TABLE]
Remark 8.4**.**
Lattice admits another basis , . There is an isomorphism
[TABLE]
8.2 Strange bosonization and odd bosonization
Representation admits fermionic, bosonic, and strange bosonic realizations; formulas are given in Propositions 5.6, 5.8 and 5.11 correspondingly. This Section is devoted to study of corresponding algebra action on . We will consider strange bosonization and its classical limit.
Let us introduce following notation (cf. with (3.2))
[TABLE]
Proposition 8.5**.**
For , ideal acts by zero on for
[TABLE]
Proof.
We will use strange bosonization of . Recall that we denote by a primitive root of unity of degree . We have
[TABLE]
Let us calculate
[TABLE]
We need to compute
[TABLE]
So
[TABLE]
Note that :\!\mathtt{e}\big{(}z^{1/n}\big{)}\mathtt{e}\big{(}\zeta z^{1/n}\big{)}\cdots\mathtt{e}\big{(}\zeta^{n-1}z^{1/n}\big{)}\!:=:\!\exp{\varphi(z)}\!:. Hence
[TABLE]
Finally note that . ∎
Remark 8.6**.**
Another way to prove Proposition 8.5 is to derive it from Proposition 7.17 (since isomorphism (8.1)). Beware inconsistency of our notation in (7.23) and (8.7). Let us rewrite (7.23)
[TABLE]
Let us consider subalgebra of Heisenberg algebra generated by for . Denote corresponding Fock module by .
Corollary 8.7**.**
There is an action of on given by
[TABLE]
Obtained representation corresponds to .
Proof.
Follows from Theorem 7.7 and Proposition 8.5 ∎
Remark 8.8**.**
Let us consider non-twisted case . Then and total sign . More accurately, the coefficient is a root of unity of degree (cf. Remark 7.21). Nevertheless, this freedom disappears if we require for (this is a standard setting for classical limit).
Example 8.9**.**
Odd bosonization is a particular case of strange bosonization for and ,
[TABLE]
Consider classical limit . It is convenient to assume and . If there exists an expansion
[TABLE]
then modes of current form ‘not -deformed’ Virasoro algebra. Note that
[TABLE]
where . Hence
[TABLE]
Or equivalently
[TABLE]
Formula (8.9) is well-known; it coincides with [48, equation (2.16)] after substitution . Let us emphasize that formula (8.8) is a -deformation of (8.9).
9 Relations on conformal blocks
9.1 Whittaker vector
In this section we define and study basic properties of Whittaker vector . We will restrict ourself to case when is irreducible.
Lemma 9.1**.**
* is irreducible if for any .*
Proof.
Follows from the proof of [16, Lemma 3.1]. ∎
In papers [41, 47] Whittaker vector is defined geometrically. We will define Whittaker vector by algebraic properties (cf. [41, Proposition 4.15]). Then we will prove that these properties define Whittaker vector uniquely up to normalization if the module is irreducible.
Definition 9.2**.**
Whittaker vector is an eigenvector of operators for with eigenvalues
[TABLE]
for ;
[TABLE]
for .We require to fix normalization (by dots we mean lower vectors).
Remark 9.3**.**
Whittaker vector is an element of graded completion of . Abusing notation, we use the same symbols for modules and their completions.
Remark 9.4**.**
Whittaker vector is an eigenvector for surprisingly big algebra. This explains why we have to consider specific eigenvalues (for general eigenvalues there is no eigenvector in corresponding representation). Theorem D.10 clarify origin of this eigenvalues.
Theorem 9.5**.**
If is irreducible, then there exists unique Whittaker vector.
One can find a proof of Theorem 9.5 in Appendix D. This statement can be considered as a part of folklore; unfortunately, we do not know a precise reference for the theorem.
Proposition 9.6**.**
Decomposition of Whittaker vector W\big{(}z^{1/n}|u\big{)}\in\mathcal{F}^{[1/n]}_{u} with respect to (8.6) is given by Whittaker vectors W\big{(}z|uq^{l_{0}},\dots,uq^{\frac{n-1}{n}+l_{n-1}}\big{)} up to normalization.
Proof.
The idea of the proof is that relations (9.1) and (9.2) for implies these relations for . Let us work out conditions (9.2) for W\big{(}z|uq^{l_{0}},\dots,uq^{\frac{n-1}{n}+l_{n-1}}\big{)}
[TABLE]
Let us calculate
[TABLE]
So W\big{(}z|uq^{l_{0}},\dots,uq^{\frac{n-1}{n}+l_{n-1}}\big{)} is defined (up to normalization) by conditions
[TABLE]
for and . Denote by generators of . Then
[TABLE]
Note that conditions (9.7)–(9.9) and conditions (9.4)–(9.6) coincide. Hence each component of W\big{(}z^{1/n}|u\big{)} also satisfy those conditions, i.e., coincide with Whittaker vector up to normalization. ∎
9.1.1 Whittaker vector for
Recall that we use notation .
Proposition 9.7**.**
We have an expansion of vector in the basis
[TABLE]
To prove the proposition we need the following lemmas.
Lemma 9.8**.**
Following vectors in coincide
[TABLE]
Proof.
Recall Cauchy identity
[TABLE]
Let us use specialization of Cauchy identity (see [36, Section 1.4, Example 2])
[TABLE]
To finish the proof let us recall that there is an identification of space of symmetric polynomials and Fock module given by and (see [34]). ∎
Remark 9.9**.**
For this specialization comes from substitution p_{k}\big{(}zq^{1/2},zq^{3/2},\dots\big{)}.
Lemma 9.10**.**
Following vectors in coincide
[TABLE]
Proof.
Recall that we have defined an operator by (3.16). Let us calculate
[TABLE]
Proposition 3.14 implies that
[TABLE]
Using Cauchy identity for another specialization
[TABLE]
we see that r.h.s. of (9.12) and (9.13) coincide. ∎
Remark 9.11**.**
For this specialization comes from substitution p_{k}\big{(}{-}zuq^{-1},-zuq^{-2},\dots\big{)}.
Proof of Proposition 9.7.
To prove the Proposition let us check that r.h.s. of (9.10) satisfy condition (9.1)–(9.3). Conditions (9.1) and (9.2) are equivalent to Lemmas 9.8 and 9.10 correspondingly. To finish the proof we note that for conditions (9.1) and (9.2) imply (9.3). ∎
Remark 9.12**.**
Note that we did not use Theorem 9.5 in the proof of Proposition 9.7. Moreover, we have proven a particular case of the Theorem for .
9.1.2 Whittaker vector and restriction on sublattice
Let us recall interpretation of decomposition (8.6) in terms of boson-fermion correspondence. One can identify . Embedding corresponded to embedding F^{na}\otimes\mathbb{C}\big{[}\mathbf{Q}_{(n)}\big{]}\subset F^{n\psi}. Hence we have decomposition
[TABLE]
We argue by construction that decomposition (8.6) correspond to (9.14).
Proposition 9.13**.**
Decomposition (9.14) identifies with for some . Moreover, partition satisfy following properties.
Hooks of are in bijection with tuples . Length of a hook corresponding to a tuple equals to . 2.
.
Proof.
The -fermion Fock space is isomorphic to tensor product . The -Heisenberg highest vectors are products . After identification of with one , these products becomes (3.11) for special . Such diagrams are called -cores.
Combinatorially boson-fermion correspondence is a correspondence between Maya diagrams and charged partitions , see, e.g., [40, Section 6.4] or [22]. Boxes of a partition correspond to pairs of white and black points in Maya diagram such that the coordinate of white point is greater than the coordinate of the black point. The hook length equals the difference between the coordinates of white and black points (cf. [40, Section 6.4]). This proves (i).
For formula (ii) see, e.g., [22, Proposition 2.30]. ∎
Lemma 9.14**.**
Let . Let us consider hooks with fixed and see Proposition 9.13).
- •
If , then possible lengths of hooks are for .
- •
If then possible lengths of hooks are for .
There are exactly such hooks of length for all possible .
For each we will use notation for product over triples satisfying following conditions. Numbers , run over with condition . If , then ; if then .
Corollary 9.15**.**
If diagram corresponds to , then
[TABLE]
Theorem 9.16**.**
Decomposition of Whittaker vector W\big{(}z^{1/n}|u^{1/n}\big{)}\in\mathcal{F}^{[1/n]}_{u^{1/n}} with respect to (8.6) is given by
[TABLE]
Proof.
Recall that according to Proposition 9.6 we just have to verify the coefficient. This coefficient can be found as coefficient of W\big{(}z^{1/n}|u^{1/n}\big{)} at highest vector of .
By Proposition 9.7, coefficient of W\big{(}z^{1/n}|u^{1/n}\big{)} at is
[TABLE]
Equality (9.15) follows from Corollary 9.15 and Proposition 9.13(ii). ∎
9.2 Shapovalov form
Definition 9.17**.**
Let , be two representations of . A pairing is called Shapovalov if .
Proposition 9.18**.**
There exists a unique Shapovalov pairing such that .
Proof.
There exists a unique pairing on Fock space such that is dual to . Since algebra is generated by modes of and , it remains to check Shapovalov property for them. Formulas (3.2) and (3.3) implies \langle v,E(z)w\rangle=-\big{\langle}F\big{(}z^{-1}\big{)}v,w\big{\rangle}. ∎
Remark 9.19**.**
Note that this pairing differs from the pairing defined in Section 3.1. More precisely, in Section 3.1 we required to be dual to , not .
Proposition 9.20**.**
\big{\langle}W\big{(}1|qu^{-1}\big{)},W(z|u)\big{\rangle}=(qz;q,q)_{\infty}.
Proof.
Formulas (9.10) and (9.11) imply
[TABLE]
Using (9.16) one can finish the proof by straightforward computation. ∎
Proposition 9.21**.**
There exists a unique Shapovalov pairing such that
[TABLE]
Proof.
There exists unique pairing satisfying (9.17). The Shapovalov property can be checked directly using (3.8)–(3.10). ∎
Proposition 9.22**.**
Shapovalov pairing for Fock modules for basis has form
[TABLE]
Proof.
Let and be Frobenius coordinates of ; analogously, and be Frobenius coordinates of . Using identification given by (3.11) we obtain
[TABLE]
Evidently, if this product is non-zero, then , , and ; this exactly means that . It remains to calculate r.h.s. of (9.18) in this case; it equals since . ∎
Definition 9.23**.**
Standard Shapovalov pairing for and is defined by
[TABLE]
Here stands for Shapovalov pairing as in Proposition 9.18.
Proposition 9.24**.**
Shapovalov pairing on restricts to for and (with respect to decomposition (8.6)). Other pairs of direct summands are orthogonal.
Proof.
Property is preserved under restriction. Since module and are irreducible, there exists unique Shapovalov pairing . So restriction of pairing coincides with the standard pairing up to multiplicative constant. The constant equals to due to Proposition 9.22 (and Proposition 9.13).
Orthogonality with all other summands also follows from Proposition 9.22 and irreducibility. ∎
Remark 9.25**.**
Let us comment on another way to prove orthogonality mentioned in Proposition 9.24. All direct summands are pairwise non-isomorphic. Hence there is no non-zero pairing for all other pairs of direct summands.
9.3 Conformal blocks
Definition 9.26**.**
Pochhammer and double Pochhammer symbols are defined by
[TABLE]
Remark 9.27**.**
Standard definition works for and any . For sufficiently small double Pochhammer symbol can be presented as
[TABLE]
The series in has non-zero radius of convergence for . Moreover the series enjoys property \big{(}u;q_{1}^{-1},q_{2}\big{)}_{\infty}=1/(q_{1}u;q_{1},q_{2})_{\infty}, hence we can define double Pochhammer symbol for any .
In particular, new definition implies \big{(}u;q,q^{-1}\big{)}_{\infty}=1/(qu;q,q)_{\infty}; this is important to compare our formulas with [5]. Below we assume .
Definition 9.28**.**
Let us define -deformed conformal block
[TABLE]
Remark 9.29**.**
AGT statement claims that function is equal to Nekrasov partition function for pure supersymmetric 5d theory. This was conjectured in [2], the proof follows from the geometric construction of the Whittaker vector given in the [41] and [47].
Theorem 9.30**.**
[TABLE]
The idea of the proof is to find two different expressions for \big{\langle}W\big{(}1|q^{1/n}\big{)},W(z^{1/n}|1)\big{\rangle} using Theorem 9.16. To do this we need to simplify after substitution . Let us concentrate on the second factor of (9.19).
Proposition 9.31**.**
[TABLE]
Proof.
Let . It is straightforward to check that
[TABLE]
Denote by for and for . Formulas (9.20)–(9.21) implies the following assertions. For
[TABLE]
For
[TABLE]
Using identities (9.22)–(9.23), we obtain
[TABLE]
To finish the proof, it remains to clarify the sign. This product already appeared as the product over all hooks. For diagram the number of hooks is . ∎
Proof of Theorem 9.30.
We will provide two different expressions for \big{\langle}W\big{(}1|q^{1/n}\big{)},\!W(z^{1/n}|1)\big{\rangle} to prove the theorem. On one hand (by Proposition 9.20)
[TABLE]
On the other hand (by Theorem 9.16 and Proposition 9.24)
[TABLE]
Note that to prove (9.25) we also used following observations: lengths of hooks in and coincides, and .
Multiplying r.h.s. of (9.24) and (9.25) by z^{\frac{1}{2}\sum\frac{i^{2}}{n^{2}}}\prod\limits_{i\neq j}\frac{1}{\big{(}q^{1+\frac{i-j}{n}};q,q\big{)}_{\infty}} we obtain
[TABLE]
Note that here we applied Proposition 9.31. ∎
Appendix A Regular product
In this section, we develop general theory of regular product. Term ‘regular product’ should be considered as an opposite to regularized (i.e., normally ordered) product.
Let be a formal power series with coefficients in for a vector space .
Definition A.1**.**
The series is called smooth if for any vector there exists such that for .
Let be a formal power series in two variables with operator coefficients. The operators acts on a vector space .
Definition A.2**.**
We will call regular if for any and for any there are only finitely many such that and .
If a current in two variables is regular one can substitute and obtain well-defined power series for any .
Let and be two smooth formal power series with operator coefficients. Recall definition of normal ordering. Denote and .
Definition A.3**.**
Normal ordered product is defined as
[TABLE]
The sign depends on parity of and in the standard way. Note that smooth formal power series in two variables is regular. Formal power series and are called local (in weaker sense) if
[TABLE]
where , and are operator valued power series.
Then one has the following OPE
[TABLE]
Proposition A.4**.**
If currents and are smooth and satisfy (A.1), then the following product is regular.
Definition A.5**.**
For define regular product
[TABLE]
From (A.2) one obtains that normally ordered product and regular product are connected by the following relation
[TABLE]
Example A.6**.**
Let us consider case of fermions , , introduced in Section 3.2. Beware, that we use notation , but (hence ). Comparing formulas (3.6)–(3.7) with Definition A.3, we conclude
[TABLE]
Using -depended normal ordering, we obtain
[TABLE]
Hence
[TABLE]
This relation was used in formula (3.9).
Example A.7**.**
Let . Then
[TABLE]
Therefore,
[TABLE]
Let us comment on deep meaning of formula (A.4). One can present algebra using currents (currents of Lie algebra type) or (currents of - algebra type). This two series of currents are connected in non-trivial way starting from . For they are related by (A.4). For general see formula (7.17) in [43].
Proposition A.8**.**
Regular product is super commutative and associative
[TABLE]
of course we assume that these regular products are well defined.
Proposition A.9**.**
Let , , and be as in Definition A.5. Then
[TABLE]
Proof.
Let be a polynomial such that following power series in two variables are regular
[TABLE]
Moreover, assume that . It is easy to see that such exists.
[TABLE]
One should differentiate this expression by application of Leibniz rule (and obtain six summands). Due to our assumptions, each of these summands is regular in the sense of Definition A.2. Hence, one can substitute to each summand separately. The proof is finished by straightforward computation. ∎
As a corollary we prove formula (7.17).
Proof of (7.17).
[TABLE]
Note that each summand is regular. Hence, we are allowed substitute to each of them separately
[TABLE]
Using Propositions A.8 and A.9, we can prove inductively that . To finish the proof, we substitute last formula into (A.5). ∎
Appendix B Serre relation
This appendix is devoted to detailed study of Serre relation.
[TABLE]
Let be a current satisfying
[TABLE]
Remark B.1**.**
Let us emphasize the difference between and . Current is a current from , but current is just a current satisfying (B.1). We need to formulate equivalent conditions to Serre relations.
Define for .
Proposition B.2**.**
There exist three currents , and such that triple commutator equals to
[TABLE]
Proof.
First of all, note that condition (B.1) is equivalent to existence of currents and such that
[TABLE]
Commutator is skew symmetric on and , hence . Now consider triple commutator . Jacobi identity and relation (B.1) imply
[TABLE]
Substituting (B.3) to (B.4), we obtain
[TABLE]
This implies that is indeed a sum of triple delta functions with some operator coefficients as in (B.2); it remains to prove proposed relations on the coefficients.
Note that triple commutator is skew symmetric on , . Also note that the sum over cyclic permutations is zero. This implies relation (B.2). ∎
Proposition B.3**.**
Serre relation for is equivalent to .
Proof.
Straightforward computation. ∎
B.1 Operator product expansion for
One can find reformulation of Serre relation in terms of OPE in [20, Section 3.3].
Proposition B.4**.**
Formal power series in three variables can be presented as sum some of regular part and singular part. Regular part is some regular power series in three variables, singular part has a form
[TABLE]
Proof.
Denote G(z_{1},z_{2},z_{3}):=\prod\limits_{i<j}(z_{i}-qz_{j})\big{(}z_{i}-q^{-1}z_{j}\big{)}E(z_{1})E(z_{2})E(z_{3}). Relation (B.1) yields to be regular. ∎
Proposition B.5**.**
Serre relation for is equivalent to condition that singular part of restricted to has no poles of order greater than .
Proof.
Note that second order pole can appear only from terms of form
[TABLE]
for . Note that these two poles can not cancel because they are at the different points . On the other hand, and . Application of Proposition B.3 completes the proof. ∎
Corollary B.6**.**
* has no poles of order greater than one. Poles may appear only at points .*
Proof.
To study we will consider OPE and substitute . Only term \big{(}z-q^{\epsilon}w_{i}\big{)}^{-1}\big{(}z-q^{\epsilon}w_{j}\big{)}^{-1}\cdots can give poles of order higher than 1 after substitution. OPE is symmetric on as a rational function. We will consider order \big{(}E(w_{i})E(z)E(w_{j})\big{)}\cdots. According to Proposition B.5, the term \big{(}z-q^{\epsilon}w_{i}\big{)}^{-1}\big{(}z-q^{\epsilon}w_{j}\big{)}^{-1}\cdots does not appear. ∎
Appendix C Homomorphism from to -algebra
This appendix is devoted to proof of Proposition 7.13. The proof is a straightforward check of relation from Proposition 2.6. The relations will be checked for operators
[TABLE]
Proposition C.1**.**
Relations (2.3), (2.4) are satisfied.
Proof.
Straightforward. ∎
Proposition C.2**.**
Currents and satisfy relation (2.5).
Proof.
It is easy to see that
[TABLE]
Thus,
[TABLE]
Proof for is analogous. ∎
Proposition C.3**.**
Currents and satisfy relation (2.6) for and .
Proof.
It is easy to see that
[TABLE]
Thus,
[TABLE]
Consequently,
[TABLE]
Denote by .
Lemma C.4**.**
\tilde{E}^{2}(z)=2\mu^{2}\exp\big{(}\frac{2}{n}\tilde{\varphi}_{-}(z)\big{)}T_{2}(z)\exp\big{(}\frac{2}{n}\tilde{\varphi}_{+}(z)\big{)}.
Proof.
[TABLE]
On the other hand, l.h.s. can be found from relation (7.8). Comparing coefficients of completes the proof. ∎
Proposition C.5**.**
* and satisfy Serre relation (2.7).*
Proof.
Using Lemma C.4, we see that
[TABLE]
Note that is regular. Proposition B.5 completes the proof of Serre relation for . Proof for is analogous. ∎
Appendix D Whittaker vector
D.1 Uniqueness of Whittaker vector
Recall that operators are defined by .
Proposition D.1**.**
Whittaker vector is annihilated by for and .
Proof.
Actually we will prove that Whittaker vector is annihilated by for . To do this we will need [43, equation (7.17)]. Let us rewrite this with respect to our notation
[TABLE]
Here denotes some combinatorially defined coefficient, which is not quite important for us. Inequality implies ; therefore . So, any summand of r.h.s. annihilates Whittaker vector. ∎
Let us denote . Denote Verma module for by (cf. Definition 7.25).
Definition D.2**.**
For each graded module let us define Shapovalov dual module . As a vector space is graded dual to . Action is defined by requirement that canonical pairing is Shapovalov.
Finally note that involution maps ideal to (maybe with different ). Hence if is a -module then so is .
Proposition D.3**.**
Let for any cf. Lemma 9.1). There is no more than one Whittaker vector .
Proof.
Denote by a subalgebra of generated by and for , and . Analogously, let be a subalgebra of generated by and for , and . Note that involution induces an involoution on which swaps and .
Consider as a -module. Whittaker vector is an eigenvector for . Hence, it is enough to show that is cocyclic for . Equivalently, we need to prove that Shapovalov dual module is cyclic for .
Fock module is isomorphic to Verma module for corresponding by Theorem 7.31. Verma module is cyclic for by an analogue of Proposition 7.26 for . ∎
D.2 Construction of Whittaker vector
Let and be a basis of .
Theorem D.4** ([1]).**
There exist homomorphisms
[TABLE]
These homomorphisms are defined uniquely up to normalization.
Remark D.5**.**
Actually operators and maps to graded completion of and correspondingly. Abusing notation, we will use the same symbol for a module and its completion. Moreover, we are going to consider a composition of such ; there appear an infinite sum as a result of such composition (a priori this sum does not make sense). We will use a calculus approach to infinite sums; below we will provide a sufficient condition for convergence of the series.
Denote by component of corresponding to . More precisely, for any in
[TABLE]
To simplify our notation we will consider particular case
[TABLE]
Note that both and are Fock modules for Heisenberg algebra generated by .
Proposition D.6** ([1]).**
Operator is defined by following explicit formulas
[TABLE]
Here sign means up to multiplication by a number. Recall
[TABLE]
Corollary D.7**.**
Heisenberg normal ordering is given by
[TABLE]
for some rational function ; here and are integer numbers.
Consider a homomorphism
[TABLE]
obtained as composition of
[TABLE]
Lemma D.8**.**
There exists a unique invariant pairing such that .
Proof.
This is equivalent to Proposition 9.18. ∎
Composition of and the pairing gives a homomorphism
[TABLE]
Let us reformulate above inductive procedure via an explicit formula
[TABLE]
As we warned in Remark D.5, operator is not a priori well defined. However, the series (appearing from the composition) converges in a domain . This assertion follows from a formula
[TABLE]
Moreover, one can consider analytic continuation of obtained function given by r.h.s. of the formula. Corollary D.7 implies that we can extend the domain to for any . Evidently, analytic continuation also enjoys intertwiner property. Hence we obtained following proposition
Proposition D.9**.**
If for any , then there is an intertwiner
[TABLE]
Denote the highest vector of by .
Theorem D.10**.**
Whittaker vector can be constructed via homomorphism as in (D.2)
[TABLE]
Proof.
Follows from (3.15). ∎
Remark D.11**.**
Recall that existence of Whittaker vector can be seen from geometric construction (see [41] and [47]).
Proof of Theorem 9.5.
Existence and uniqueness follows from Theorem D.10 and Proposition D.3 correspondingly. ∎
D.3 Whittaker vector for algebra
Let us define coefficient by (cf. (D.1))
[TABLE]
Also recall that for
[TABLE]
Definition D.12**.**
For Whittaker vector with respect to is an eigenvector for (for and ) with eigenvalues given by
[TABLE]
We require to fix normalization (by dots we mean lower vectors).
One can find notion of Whittaker vector for in the literature (see [46]). In this section we will explain connection between notion of Whittaker vector and Whittaker vector for (see Definition 9.2). Our plan to explain this connection is as follows. First we define Whittaker vector with respect to (we denote it by ). Then we will see, that on the one hand, the vector is connected with ; on the other hand it is connected with .
Recall, that ; ideal annihilates . Hence is a representation of .
Definition D.13**.**
For Whittaker vector is a vector belonging to and satisfying following conditions
[TABLE]
We require to fix normalization (by dots we mean lower vectors).
Recall that with respect algebra identification .
Lemma D.14**.**
Vector satisfies properties of iff it is .
Proof.
Note that
[TABLE]
Moreover has only terms of positive degree in . Hence we expressed action of via for . Therefore we have proven that satisfies property of .
The implication in opposite direction is analogous. ∎
Proposition D.15**.**
There exists at most one vector .
Proof.
Analogous to proof of Proposition D.3. The only difference is that we consider a different character of subalgebra . ∎
We need to generalize notion of Whittaker vector for to compare it with .
Definition D.16**.**
For any , Whittaker vector is an eigenvector of operators for and . More precisely,
[TABLE]
for ,
[TABLE]
for and .We require to fix normalization (by dots we mean lower vectors).
Recall that we have defined operator by (3.16). By Proposition 3.14 the operator enjoys intertwiner property . Denote . Note that also enjoys intertwiner property (here denotes the homomorphism of the representation ).
Proposition D.17**.**
.
Corollary D.18**.**
There exists unique if .
Proposition D.19**.**
There exists unique vector . Moreover, .
Proof.
We already know uniqueness of and existence of from Proposition D.15 and Corollary D.18 correspondingly. So it is sufficient to show that satisfies properties of . Last assertion follows from formula (D.1) (also see (D.3)). ∎
Theorem D.20**.**
There exists unique vector . Moreover, .
Proof.
Follows from Lemma D.14 and Proposition D.19. ∎
Acknowledgments
We are grateful to B. Feigin, P. Gavrylenko, E. Gorsky, A. Neguţ, J. Shiraishi, for interest to our work and discussions. The work is partially supported by Russian Foundation of Basic Research under grant mol_a_ved 18-31-20062 and by the HSE University Basic Research Program jointly with Russian Academic Excellence Project ‘5-100’. R.G. was also supported in part by Young Russian Mathematics award. The results of Section 9 are obtained under the support of the Russian Science Foundation under grant 19-11-00275.
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