Twisted Representation of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks

We study certain representations of quantum toroidal $\mathfrak{gl}_1$ algebra for $q=t$. We construct explicit bosonization of the Fock modules $\mathcal{F}_u^{(n',n)}$ with nontrivial slope $n'/n$. As a vector space, it is naturally identified with the basic level 1 representation of affine $\mathfrak{gl}_n$. We also study twisted $W$-algebras of $\mathfrak{sl}_n$ acting on these Fock modules. As an application, we prove the relation on $q$-deformed conformal blocks which was conjectured in the study of $q$-deformation of isomonodromy/CFT correspondence.


Introduction
Toroidal algebra Representation theory of quantum toroidal algebras is actively developed in recent years. This theory has numerous applications, including geometric representation theory and AGT relation [Neg18], topological strings [AFS12], integrable systems, knot theory [GN15], and combinatorics [CM18].
In this paper we consider only quantum toroidal gl 1 algebra, we denote it by U q,t (gl 1 ). The corresponding algebra depends on two parameters q, t and has PBW generators E k,l , (k, l) ∈ Z 2 and central generators c ,c [BS12]. In the main part of the text we consider only the case q = t, where toroidal algebra becomes universal enveloping of the Lie algebra with these generators E k,l , c , c and relation [E k,l , E r,s ] = (q (sk−lr)/2 − q (lr−sk)/2 )E k+r,l+s + δ k,−r δ l,−s (c k + cl). (1.1) We denote this Lie algebra by Diff q since there is a homomorphism from this algebra to the algebra of q-difference operators generated by D, x with relation Dx = qxD; namely E k,l → q kl/2 x l D k . There is another presentation of the algebra Diff q (and more generally U q,t (gl 1 )) using Chevalley generators E(z) = k∈Z E 1,k z −k , F (z) = k∈Z E −1,k z −k , H(z) = k =0 E 0,k z −k , see e.g. [Tsy17].
In this paper we deal with Fock representations of Diff q ; to be more precise there is a family F u of Fock modules, depending on parameter u (see Proposition 3.1 for a construction of F u ). They are just Fock representations of the Heisenberg algebra generated by E 0,k . Images of E(z) and F (z) are vertex operators. Such construction of representation is usually called bosonization.
It was shown in [FHH + 09], [Neg18] that the image of toroidal algebra U q,t (gl 1 ) in the endomorphisms of tensor product of n Fock modules is deformed W -algebra for gl n . There is so called conformal limit q, t → 1 in which deformed W -algebras go to vertex algebras. These vertex algebras are tensor products of Heisenberg algebra and W -algebras of sl n . In the q = t case, the central charge of the corresponding W -algebra of sl n is equal to n − 1. These W -algebras appear in the study of isomonodromy/CFT correspondence [GIL12], [GM16]. This is one of the motivations of our paper.
The q-deformation of the isomonodromy/CFT correspondence was proposed in [BS17b], [BGT19], [JNS17]. The main statement is an explicit formula for the q-isomonodromic tau function as an infinite sum of conformal blocks for deformed W -algebras with q = t. In general these tau functions are complicated, but there are special cases (corresponding to algebraic solutions) where these tau functions are very simple ([BS17b], [BGM19]). These cases should correspond to special representations of q-deformed W -algebras. The construction of such representation is one of the purposes of this paper.
Twisted Fock modules There is a natural action of SL(2, Z) on Diff q . We will parametrize σ ∈ SL(2, Z) by σ = m m n n .
(1.2) Then σ acts as σ(E k,l ) = E m k+ml, n k+nl , σ(c ) = m c + n c, σ(c) = mc + nc. (1.3) For any Diff q module M and σ ∈ SL(2, Z) denote by M σ module twisted by the automorphism σ (see Definition 2.3). The twisted Fock modules depend only on n and n (up to isomorphism). These numbers are values in F σ u of the central generators c and c correspondingly. Therefore we will also use notation F (n ,n) u for F σ u . Twisted Fock modules F σ u (for generic q, t) were used for example in [AFS12] and [GN17].
In the Section 4 we construct explicit bosonization of the twisted Fock modules F σ u for q = t. Actually, we give three constructions: first in terms of n-fermions (see Theorem 4.1), second in terms of n-bosons (see Theorem 4.2) and third in terms of one twisted boson (see Theorem 4.3) (here, for simplicity, we assume that n > 0). In other terms the twisted Fock module will be identified with the basic module for gl n ; these two bosonizations corresponds to homogeneous [FK81] and principal [LW78], [KKLW81] constructions.
The construction of the bosonization is nontrivial since it is given in terms of Chevalley generators (the SL(2, Z) action is not easy to describe in terms of Chevalley generators). The appearance of affine gl n is in agreement with Gorsky-Negut , conjecture [GN17]. More precisely, it was conjectured in [GN17] that there exists an action (with certain properties) of U p 1/2 ( gl n ) on F σ u for p = q/t = 1; we expect that this is p-deformation of the gl n -action constructed in this paper.
It is instructive to look to the formulas in the simplest examples. Also, for simplicity, we give here only formulas for E(z). Here we introduce notation in a sloppy way, see details in Sections 3 and 4.
We present two different proofs of the Theorems 4.1, 4.2, and 4.3. The first one is given in Section 5 and is based on the following idea. For any full rank sublattice Λ ∈ Z 2 of index n we have a subalgebra Diff Λ q 1/n ⊂ Diff q 1/n which is spanned by E a,b for (a, b) ∈ Λ and central elements c, c . The algebra Diff Λ q 1/n is isomorphic to Diff q , the isomorphism depends on choice of positively oriented basis v 1 , v 2 in Λ. Denote this isomorphism by φ v 1 ,v 2 .
If the basis v 1 , v 2 is chosen such that v 1 = (N, 0), v 2 = (R, d), then the restriction of the Fock module F u on φ v 1 ,v 2 (Diff q ) is isomorphic to the sum of tensor products of the Fock modules F uq rl 0 ⊗ · · · ⊗ F uq r( α n +lα) ⊗ · · · ⊗ F uq r ( d−1 d +l d−1) (1.8) where r = gcd(N, R) and Q (d) = {(l 0 , . . . , l d−1 ) ∈ Z d | l i = 0}. If we chose bases w 1 , w 2 in Λ which differs from v 1 , v 2 by σ ∈ SL(2, Z) we get analogue of decomposition (1.8) with RHS given by a sum of tensor product of twisted Fock modules. For the basis w 1 = (r, n tw ), w 2 = (0, n) we write formulas for Chevalley generators of Diff q = Diff Λ q 1/n using either initial fermion or initial boson for F u . Applying this for the lattices with d = 1, we get Theorems 4.1, 4.2, 4.3.
The second proof of these theorems is based on the semi-infinite construction. Let V u denotes representation of the algebra Diff q in a vector space with basis x k−α for k ∈ Z where Diff q acts as q-difference operators (see Definition 3.1). This representation is called vector (or evaluation) representation; the parameter u is equal to q −α . The Fock module F u is isomorphic to Λ ∞/2+0 (V u ) ⊂ Λ ∞/2 (V u ). After the twist we get semi-infinite construction of F σ u ⊂ (Λ ∞/2 V u ) σ = Λ ∞/2 (V σ u ). Note that conjecturally semi-infinite construction of F σ u can be generalized for q = t (cf. [FFJ + 11a]).
Twisted W algebras Denote by Diff 0 q subalgebra of Diff q generated by c and E a,b , for a 0. There is an another set of generators E k [j] of the completion of the U (Diff 0 q ), defined by the formula j∈Z E k [j]z −j = (E(z)) k (see Appendix A for the definition of the power of E(z)). The currents H(z) and E k (z) for k ∈ Z >0 satisfy relations of the q-deformed W -algebra of gl ∞ (see [Neg18]). We denote this algebra by W q (gl ∞ ).
There exists another description of this story using q-deformed W -algebra of sl n introduced in [FF96]. Define T k [j] by the formula The generators T k [j] are elements of a localization of the completion of U (Diff 0 q ). These generators commute with H i and satisfy certain quadratic relations. The algebra generated by There is an ideal in W q (sl ∞ ) which acts by zero on any tensor product F u 1 ⊗ · · · ⊗ F u d , This ideal contains relations c = d, T d (z) = 1, and T d+k (z) = 0 for k > 0. The quotient is a standard W -algebra W q (sl d ) [FF96] (see also Definition 7.1). We have a relation W q (gl d ) = W q (sl d ) ⊗ U (Heis), where Heis is Heisenberg algebra generated by E 0,j .
In the case of product of twisted Fock modules F σ u 1 ⊗ · · · ⊗ F σ u d the situation is similar. The corresponding ideal contains relations T nd (z) = z n d , T nd+k (z) = 0 for k > 0. We present the quotient in terms of generators T 1 (z), . . . , T nd (z) and relations (this is Theorem 7.1). We call the algebra with such generators and relations by twisted W -algebra W q (sl nd , n d) ; see Definition 7.2. 1 The quadratic relations in the algebra W q (sl nd , n d) are the same as in untwisted case (see eq. (7.2)-(7.3)), the only difference lies in relation T nd (z) = z n d .
The algebra W q (sl nd , n d) is graded, with deg T k [j] = j + n k n . Rename generators by T tw k [r] = T k [r − n k n ], for r ∈ n k n + Z. Presentations of the algebra W q (sl nd , n d) in terms of generators T tw k [r] and the algebra W q (sl nd ) is terms of generators T k [r] are given by the same formulas, the only difference is the region of r. Heuristically, one can think that W q (sl nd , n d) is the same algebra as W q (sl nd ) but with currents having nontrivial monodromy around zero.
Example 1.5. One can also use embedding Diff Λ q 1/n ⊂ Diff q 1/n in order to construct bosonization of W -algebras. Namely one can take representation of Diff q 1/n with known bosonization and then express W -algebra related to Diff q = Diff Λ q 1/n in terms of these bosons. For example, consider Λ generated by v 1 = e 1 , v 2 = 2e 2 and Fock representations F u 1/2 of Diff q 1/2 . One can show (for example, using (1.8)) that W q (gl ∞ ) algebra related to Diff q ∼ = Diff Λ q 1/2 acts on F u 1/2 through quotient W q (gl 2 ). Therefore, we get an odd bosonization of non-twisted q-deformed Virasoro algebra W q (sl 2 ) (1.17) Here J r are odd part of the initial boson for F u . The even part of the boson disappears in this formula since it belongs to Heis ⊂ Diff Λ q 1/2 . It follows from the decomposition (1.8) that formula (1.17) gives bosonization of special representation W q (sl 2 ), to be more precise a direct sum of Fock modules (defined by (1.14)) with particular parameters u = q l−1/4 for l ∈ Z.
Whittaker vectors and relations on conformal blocks As an application in Section 9 we prove the following identity Here the lattice Q (n) is as above, (u; q, q) ∞ = ∞ i,j=0 (1 − q i+j u). The function Z(u 1 , . . . , u n ; z) is a Whittaker limit of conformal block. By AGT relation it equals to the Nekrasov partition function. We recall the definition of Z(u 1 , . . . , u n ; z) below.
The relation (1.18) was conjectured in [BGM19] in the framework q-isomonodromy/CFT correspondence. The right side of (1.18) is specialization of generic conjectural formula [BGM19, eq. (3.6)] for the tau function of deautonomized discrete flow in Toda system. The left side of (1.18) is a tau function corresponding to the algebraic solution, see [BGM19, eq. (3.11)].
Let us recall the definition of Z(u 1 , . . . , u n ; z). Whittaker vector W (z|u 1 , . . . , u N ) is a vector in a completion of F u 1 ⊗ . . . ⊗ F un , which is an eigenvector of E a,b for N b a 0 with certain eigenvalues depending on z, see Definition 9.1. Such vector exists and unique for generic values of u 1 , . . . , u n . This property looks to be a part of folklore, we give a proof of it in Appendix D. The proof is essentially based on the results of [Neg18], [Neg17]. The function Z is proportional to a Shapovalov pairing of two Whittaker vectors Z(u 1 , . . . , u n ; z) = z We give a proof of (1.18) using decomposition (1.8). We consider Whittaker vector W (z|1) for the algebra Diff q 1/n . Its Shapovalov pairing gives the left side of the relation (1.18). On the other hand, we prove that its restriction to summands F q l 0 ⊗ · · · ⊗ F q n−1 n +l n−1 is a Whittaker vector for the algebra Diff q . So taking the Shapovalov pairing we get the right side of the relation (1.18).
In the conformal limit q → 1 the analogue of the relation (1.18) in case n = 2 was proven in [BS17a] by a similar method. The conformal limit of the decomposition (1.8) was studied in [BGM18].
Plan of the paper. The paper is organized as follows.
In Section 2 we recall basic definitions and properties on algebra Diff q .
In Section 3 we recall basic constructions of Fock module F u . In Section 4 we present three constructions of twisted Fock module F σ u : fermionic construction in Theorem 4.1, bosonic construction in Theorem 4.2, and strange bosonic construction in Theorem 4.3.
In Section 5 we study restriction of Fock module to a subalgebra Diff Λ q . Using these restrictions we prove Theorems 4.1, 4.2, 4.3.
In Section 6 we give an independent proof of Theorem 4.1 using semi-infinite construction. In Section 7 we study twisted q-deformed W -algebras. We define W q (sl n , n tw ) by generators and relations. Then we show in Theorem 7.1 that tensor product W q (sl n , n tw ) ⊗ U (Heis) is isomorphic to the certain quotient of U (Diff q ); we denote this quotient by W q (gl n , n tw ). We show that W q (sl nd , n d) acts on the tensor product of twisted Fock modules F σ u 1 ⊗ · · · ⊗ F σ u d . At the end of the section we study relation between these modules and Verma modules for W q (gl nd , n d) and W q (sl nd , n d)..
In Section 8 we prove decomposition (1.8). Then we study strange bosonization of W -algebra modules arising from the restriction of Fock module on Diff Λ q . In Section 9 we recall definitions and properties of Whittaker vector, Shapovalov pairing, and conformal blocks. Then we prove (1.18), see Theorem 9.3.
In Appendix A we give definition and necessary properties of regular product of currents A(z)B(az) for a ∈ C.
Appendices B and C consist of calculations which are used in Section 7. In Appendix D we study Whittaker vector for Diff q in the completion of the tensor product F u 1 ⊗ . . . ⊗ F un . We prove its existence and uniqueness (we use this in Section 9). To proof existence we present a construction of Whittaker vector via intertwiner operator from [AFS12]. We also relate this Whittaker vector to the Whittaker vector of W q (sl n ) introduced in [Tak10].
Remark 2.1. Note that vector subspace of Diff A q spanned by x l D k (for (l, k) = (0, 0)) is closed under commutation i.e. has a natural structure of Lie algebra (denote this Lie algebra by Diff L q ). Consider a basis of this Lie algebra E k,l := q kl/2 x l D k . Finally, Diff q is a central extension of Diff L q by twodimensional abelian Lie algebra spanned by c and c .

SL 2 (Z) action
In this section we will define action SL 2 (Z) on Diff q . Let σ be an element of SL 2 (Z) corresponding to a matrix σ = m m n n .
(2.2) Then σ acts as follows σ(E k,l ) = E m k+ml, n k+nl , σ(c ) = m c + n c, σ(c) = mc + nc. We will refer to M σ as twisted representation. More precisely, M σ is the representation M , twisted by σ.

Chevalley generators and relations
Lie algebra Diff q is generated by E k := E 1,k , F k := E −1,k and H k := E 0,k . We will call them Chevalley generators of Diff q . Define following currents (formal power series with coefficients in Diff q ) (2.5) Let us also define formal delta function Proposition 2.2. Lie algebra Diff q is presented by generators E k , F k (for all k ∈ Z), H l (for l ∈ Z\{0}), c, c and relations (2.12)

Fock module
In this section we review basic constructions of representations of Diff q with c = 1 and c = 0. These construction were studied in [GKL92].

Free boson realization
Introduce Heisenberg algebra generated by a k (for k ∈ Z) with relation [a k , a l ] = kδ k+l,0 . Consider Fock module F a α generated by |α such that a k |α = 0 for k > 0, a 0 |α = α|α .
Proposition 3.1. The following formulas define an action of Diff q on F a α c → 1, c → 0, H k → a k ; (3.1) We will denote this representation by F u .
Remark 3.1. Note that α does not appear in formulas (3.1)-(3.3). But we will need operator a 0 later (see proof of Proposition 3.6) for boson-fermion correspondence. Heuristically, one can think that u = q −α .
Remark 3.2 (on our notation). In this paper we consider several algebras and their action on corresponding Fock modules. We choose following notation. All these representations are denoted by letter F (for Fock) with some superscript to mention an algebra. Since Diff q is the most important algebra in our paper, we use no superscript for it's representation. Also, let us remark that we consider several copies of Heisenberg algebra. To distinguish their Fock modules we write letter for generators as a superscript.
Standard bilinear form on F a α is defined by following conditions: operator a −k is dual of a k , pairing of |α with itself equals to 1. We will use bra-ket notation for this scalar product. For operator A we denote by α|A|α scalar product of A|α with |α .
Proof. Consider current It is easy to verify that [a k , T (z)] = 0. Since F α is irreducible, T (z) = f (z) for some formal power series f (z) with C-coefficients. On the other hand f (z) = α|E(z)|α = u 1−q . This implies (3.2). Proof of (3.3) is analogous.
Proposition 3.3. Denote E l (z) = E l,k z −k . The action of E l (z) on Fock representation F u is given by Proof. The commutation relation (2.1) implies that the formula (3.4) holds up to a pre-exponential factor. Also we see from (2.1) that The factor can be found inductively from (3.5).

Semi-Infinite construction
Definition 3.1. Evaluation representation V u of algebra Diff q is a vector space with basis x k for k ∈ Z and action E a,b x k = u a q ab 2 +ak x k+b ; c = c = 0.
(3.15) Remark 3.4. Associative algebra Diff A q acts on V u . Representation of Diff q is obtained via evaluation homomorphism ev : Diff q → Diff A q . Remark 3.5. Informally, one can consider x k ∈ V u as x k−α for u = q −α . Define action of Diff A q as follows. Generator x acts by multiplication and Dx k−α = q k−α x k−α = uq k x k−α . However q −α is not well defined for arbitrary complex α. So we consider u as a parameter of representation instead of α.
Let us consider semi-infinite exterior power of evaluation representation Λ ∞/2 V u . It is spanned by where λ is a Young diagram and l ∈ Z. Let p 1 > . . . > p i and q 1 > . . . > q i be Frobenius coordinates of λ.
Proposition 3.6. There is an isomorphism of Diff q modules M u ∼ = l∈Z F q l u . Submodule F q l u is spanned by |λ, l .
Proof. Recall ordinary boson-fermion correspondence (see [KR87]). Coefficients of a(z) = n a n z −n−1 =: ψ(z)ψ * (z) : (0) (3.17) are indeed generators of Heisenberg algebra. Moreover F ψ = ⊕ l∈Z F a −l . The highest vector of F a −l is |l (in particular, a 0 |l = −l|l ). Note that this is decomposition of Diff q modules as well. Also note that Therefore one can use Proposition 3.2 for each summand F a −l .
There is a basis in the Fock module F u given by semi-infinite monomials (3.18) To write action of Diff q in this basis let us remind standard notation. Let l(λ) be the number of non-zero rows. We will write s = (i, j) for jth box in ith row (i.e. j λ i ). Content of a box c(s) := i − j. For diagram µ ⊂ λ define skew Young diagram λ\µ is set of boxes λ which are not in µ. Ribbon is a skew Young diagram without 2 × 2 squares. Height ht(λ\µ) of a ribbon is one less than the number of its rows.
Proposition 3.7. The action of Diff q on F u is given by the following formulas In particular, (3.22) Let us introduce notation c(λ) = s∈λ c(s). Define an operator I τ ∈ End(F u ) by formula Corollary 3.9. F τ u ∼ = F u for τ = ( 1 1 0 1 ) Remark 3.6. Also Corollary 3.9 follows from Proposition 3.2: we will use this approach to proof Proposition 5.2.
Corollary 3.10. Twisted representation F σ u is determined up to isomorphism by n and n . Proof. Corollary 3.9 implies that F τ k σ u ∼ = F σ u . Note that For fixed n and n all possible choices of m and m appear for appropriate k.

Explicit formulas for twisted representation
In this section we provide three explicit constructions of twisted Fock module F σ u for σ = m m n n . (4.1) 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.

Fermionic construction
We need to consider Z/2Z graded nth tensor power of Clifford algebra defined above. More precisely, consider an algebra, generated by ψ (a) [i] and ψ * (b) [j] for i, j ∈ Z; a, b = 0, . . . , n − 1 subject to relations Consider currents Consider a module F nψ with a cyclic vector |l 0 , . . . , l n−1 and relation The module F nψ does not depend on l 0 , . . . , l n−1 . The isomorphism can be seen from formulas Obtained module is isomorphic to M σ u .
Since F σ u ⊂ M σ u , we have obtained fermionic construction for F σ u .

Bosonic construction
Let us consider nth tensor power of Heisenberg algebra. More precisely, this algebra is generated by However this operator will not act on our representation. We will use Q b as a formal symbol. Our final answer will consist only e Q b , but not Q b without exponent.
We need a notion of a normally ordered exponent : exp(. . .) :. The argument of normally ordered exponent is a linear combination of a b [i] and Q b . Let a + , a − , a 0 , and Q denote a linear combination (4.14) Also note, that a 0 will have coefficient log z. We shall understand it formally; action of the operator exp(a b [0] log z) = z a b [0] is well defined since in representation to be considered below, a b [0] acts as multiplication by integer number at each Fock module.
Let Q (n) be a lattice with basis Q 0 −Q 1 , . . . , Q n−2 −Q n−1 . Consider group algebra C[Q (n) ]. This algebra is spanned by e λ for λ = i λ i Q i ∈ Q (n) . Let us define action of commutative algebra generated by Finally, we can consider F na ⊗C[Q (n) ] as representation of whole Heisenberg algebra as follows: a b [i] for i = 0 acts on the first factor, a b [0] acts on the second factor. Also, Theorem 4.2. There is an action of Diff q on F na ⊗ C[Q (n) ] defined by formulas Obtained representation is isomorphic to F σ u .

Strange Bosonic construction
We will use notation of Section 3.1. Let ζ be a nth primitive root of unity, e.g. ζ = e 2πi n .
Theorem 4.3. There is an action of Diff q on F a α defined by formulas.
Obtained representation is isomorphic to F σ u .
As before the representation does not depend on α, see Remark 3.1.

Sublattices and subalgebras
Consider a full rank sublattice Λ ⊂ Z 2 of index n (i.e. Z 2 /Λ is a finite group of order n). Let us define a Lie subalgebra Diff Λ q ⊂ Diff q which is spanned by E a,b for (a, b) ∈ Λ and central elements c, c .
Slightly abusing notation, denote Fock representation of Diff q n by F Recall that character of Z-graded module is generating function of dimensions of graded components. Then character of Fock module is ch . Consider a subalgebra Heis 0 in Diff q n spanned by E 0,k and c. Note that Heis 0 is isomorphic to it coincides with ch F u . This implies that F u | φv 1 ,v 2 (Diff q n ) restricted to Heis 0 is isomorphic to Heis 0 Fock module. To finish the proof, we use Propositions 3.2 and 3.3.

Twisted Fock vs restricted Fock
From now on we change q → q 1/n . Our goal is to construct an action of Diff q on Fock module twisted by σ ∈ SL 2 (Z) as in (4.1) for n = 0. Consider a sublattice Λ σ ⊂ Z 2 spanned by v 1 = (n, 0) and v 2 = (−m, 1). Consider another basis of Λ σ obtained by σ w 1 = m v 1 + n v 2 = (m n − n m, n ) = (1, n ) (5.5) Remark 5.1. Construction of this sublattice Λ σ ⊂ Z 2 naturally appears if you require σ to be a transition matrix from v i to w i and assume v 1 = (n, 0), w 2 = (0, n).
Denote Fock module of Diff q 1/n by F On the other hand, relations (5.5) and (5.6) yield that σ is transition matrix from w 1 , w 2 to v 1 , v 2 .
Theorem 5.1 combined with results from Section 3 enables us to find explicit formulas for action on F σ u . We will do this below.
Proposition 5.4. Formulas below define an action of Diff q on F nψ Remark 5.2. Below we will substitute n tw = n to prove Theorem 4.1. However, Proposition 5.4 is more general, then it is necessary for the proof, since we do not assume here that gcd(n, n tw ) = 1. We will need case of arbitrary n tw in Section 8.
Proof. We use notation E(z), F (z) and H(z) for Chevalley generators of Diff q 1/n . Generators of Diff q ∼ = Diff Λ q 1/n (identified by ϕ w 1 ,w 2 ) will be denoted by E tw (z), F tw (z) and H tw (z). Let us write identification φ w 1 ,w 2 explicitly for Chevalley generators Let us consider currents ψ (a) (z) and ψ * (b) (z) for a, b = 0, 1, . . . , n − 1. These currents are defined by following equations Let us denote modes of ψ (a) (z) and ψ * (b) (z) as in equation (4.4). It is easy to see that those modes satisfy Clifford algebra relations (4.2), (4.3). So we have identified Clifford algebra and nth power of Clifford algebra. This leads to identification F ψ = F nψ . Substituting (5.16) into (5.8) and (5.9), we obtain For technical reasons we need to treat cases n tw = 0 and n tw = 0 separately. Let us first consider case n tw = 0. Using formulas (5.13)-(5.15), we see that For n tw = 0 we obtain This can be rewritten as Note that l-dependent normal ordering is defined it terms of ψ i and ψ * j . One can check (cf. (A.6)) Proof of Theorem 4.1. Follows from Theorem 5.1 and Proposition 5.4.

Bosonic construction via sublattices
here a,b = r (−1) ar[0] (we consider product over such r that a − 1 r b for a > b and b − 1 r a for b > a).
Proof Proposition 5.5. We need an upgraded version boson-fermion correspondence for the proof. Namely, there is an action of nth tensor power of Heisenberg algebra on F nψ given by Proof. One should substitute (5.32) into fermionic formulas (5.10)-(5.12).
Lemma 5.7 implies that the identification of vector spaces F ψ = F nψ leads to identification of subspaces Let us package identifications of vector subspaces into a commutative diagram Proposition 5.4 states that formulas (5.10)-(5.12) gives an action of Diff q with respect to identification of bottom line of the diagram. Therefore, Lemma 5.6 implies that formulas (5.27)-(5.29) describes the action of Diff q with respect to identification of top line of the diagram.
Proof of Theorem 4.2. Follows from Theorem 5.1 and Proposition 5.5.

Strange bosonic construction via sublattices
Proposition 5.8. There is an action of Diff q on F a α defined by formulas.
Proof of Theorem 4.3. Follows from Theorem 5.1 and Proposition 5.8.

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 σ = m m n n Twisted evaluation representation Let e a,b be a matrix unit (all entries are 0 except for one cell, where it is 1; this cell is in bth column and ath row). Consider a homomorphism t u,σ : Proof. Consider a basis v l := q ml 2 2n u ml n x l of evaluation representation C[x, x −1 ] σ . Action with respect to this basis looks like Let a, b = 0, . . . , n − 1 such that l = nj + b and a ≡ b + n mod n. Substituting l = nj + b into (6.5) we obtain Then formula (6.7) will be rewritten To be compared with formula (6.2) this proves the proposition for E 1,k . Prove for E −1,k is analogous. For E 0,k proposition is obvious from (6.4).
Semi-infinite construction. To apply semi-infinite construction we need to pass from associative algebras to Lie algebras. 0) and a, b = 0, . . . , n − 1), c and c . Elements c and c are central. All other commutators are given by Proposition 6.2. There is an action of Diff q (gl n ) on F nψ given by formulas c → 1; c → 0 (6.10) Proof of Theorem 4.1. According to Proposition 3.5, Therefore, Propositions 6.1 and 6.2 imply Theorem 4.1 7 q-W -Algebras

Definitions
Topological algebras and completions In this section we will work with topological algebras. Let us define topological algebra appearing as a completion of Diff q . It is given by projective limit of U (Diff q )/J k where J k is the left ideal generated by non-commutative polynomials in E j 1 ,j 2 of degree −k (with respect to grading deg E j 1 ,j 2 = −j 2 ). Although each U (Diff q )/J k 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 Diff q and its completion and so on.
Remark 7.1. There are different approaches to definition of q-W -algebra. For example, in [FF96] algebra W q,p (sl n ) was defined via bosonization. The currents T k (z) satisfy relation [FF96, Thm. 2] where f k,n (x) = (x|p m−1 q, p m q −1 , p n , p n−1 ; p n ) (x|p m−1 , p m , p n−1 q, p n q −1 ; p n ) . (7.7) One can check that limit p → 1 gives relation (7.4). However [FF96] do not provide presentation of W q,p (sl n ) in terms of generators and relations.
In the paper [Neg18] relation [Neg18, (2.62)] defines algebra W q,p (gl n ) which (non-essentially) differs from W q,p (sl n ) mentioned above (and from W q (sl n ) defined above).

Twisted q-W -algebras
Twisted q-W -algebra depends on remainder of n tw modulo n. If n tw = 0, then we get definition of non-twisted q-W -algebra from last section. One can find definition of W q,p (sl 2 , 1) in [Shi04, (37)-(38)].
Definition 7.2. Algebra W q (sl n , n tw ) is generated by T tw k [r] for r ∈ n tw k/n + Z and k = 1, . . . , n − 1. It is convenient to add T tw 0 [r] = T tw n [r] = δ r,0 . The defining relations are (7.9) Let us rewrite relations (7.8)-(7.9) in the current form. Define currents Remark 7.2. In non-twisted case we have relations (7.4) and (7.5) for currents T k (z). In twisted case we have the same relations, but for two different sets of currents T k (z) and T • k (z). One should also keep in mind (7.12).
7.2 Connection of W q (sl n , n tw ) with Diff q Connection between W q (sl n ) and Diff q is known (see [FHH + 09, Prop. 2.14] or [Neg18, Prop. 2.25]). In this section we generalize it for arbitrary n tw .
Let Heis be a Heisenberg algebra generated byH j with relation [H i ,H j ] = niδ i+j,0 . We will prove that there is a surjective homomorphism Diff q W q (sl n , n tw ) ⊗ U (Heis). Secretly, generators H j are mapped toH j under the homomorphism. Let us introduce a notation to describe this homomorphism more precisely. Define (7.16) Also, let introduce notation Note thatT k (z) commute with H j . Let J µ,n,ntw be two sided ideal in Diff q generated by c − n, c − n tw andT n (z) − µ n z ntw (here µ ∈ C\{0}). Parameter µ is not essential since automorphism E a,b → µ −a E a,b maps J µ,n,ntw to J 1,n,ntw . So we will abbreviate J n,ntw = J µ,n,ntw .
Theorem 7.1. There is an algebra isomorphisms S : W q (sl n , n tw ) ⊗ U (Heis) The map P in opposite direction is given by H j →H j ; c → n; c → n tw ; (7.22) The rest of this section is devoted to proof of Theorem 7.1. First of all, we will prove that formula (7.21) indeed defines a homomorphism S : W q (sl n , n tw ) ⊗ U (Heis) → U (Diff q )/J n,ntw (see Proposition 7.7). Then we prove that formulas (7.22)-(7.24) defines a homomorphism in opposite direction (see Proposition 7.8). Finally, we note that maps P and S are mutually inverse.
Proof. Let us define power series in two variables According Corollary B.5, E (k+1) (z, w) is regular in sense of Definition A.2. Following relations follows from results of Appendix A More precisely, (7.26)-(7.27) easily follows from Propositions A.2. One can find a proof of (7.28) at the end of Appendix A.
Proposition 7.7. Formula (7.21) defines a homomorphism S from W q (sl n , n tw ) ⊗ U (Heis) to the algebra U (Diff q )/J n,ntw .
Proof. Evidently, H j andT k (z) commute, and H j form a Heisenberg algebra. We only have to check that 1 µ kTk (z) form W q (sl n , n tw ) algebra. Relation of W q (sl n , n tw ) algebra follows from Propositions 7.3 and 7.6.
Proof. Let us check that these formulas define morphism from Diff q . According to Proposition 2.2, it is enough to proof relations (2.8)-(2.13). It is done in Appendix C. Evidently, P annihilates J n,ntw .
Proposition 7.9. Maps P and S are mutually inverse.
Proof. Let us to prove PS = id Wq(sln,ntw)⊗U (Heis) first. The algebra W q (sl n , n tw ) is generated by modes of T 1 (z). Hence it is sufficient to check PS(H n ) =H n and PS T 1 (z) = T 1 (z). Both of them are straightforward.
The algebra U (Diff q )/J n,ntw is generated by modes of E(z) and F (z). Evidently, SP E(z) = E(z). Proposition 7.5 implies SP F (z) = F (z).
Proof of Theorem 7.1. Follows from Proposition 7.7, 7.8 and 7.9 7.3 Bosonization of W q (sl n , n tw ) Let σ be as in (4.1). Corollary 3.10 states that representation F σ u actually does not depend on m and m; it is determined by n and n. Let us denote the representation by F (n ,n) u .

Fock representation via Diff q
In this section we will discuss connection of twisted q-W algebras and twisted representations F (n ,n) u . Lemma 7.10. In representation F (n ,n) u operatorT n (z) acts by −q −1/2 uz n 1 (q 1/2 −q −1/2 ) n .
Proof. We will use formula (4.11) to calculate E n (z).
Theorem 7.2. There is an action of W(sl nd , n d) ⊗ U (Heis) on F (n ,n) u 1 ⊗ · · · ⊗ F (n ,n) u d such that action of Diff q factors through W(sl nd , n d) ⊗ U (Heis).
Proof. According to Proposition 7.12, algebra U (Diff q )/J nd,n d acts on F (n ,n) u 1 ⊗ · · · ⊗ F (n ,n) u d . By Theorem 7.1, algebra U (Diff q )/J nd,n d is isomorphic to W(sl nd , n d) ⊗ U (Heis).
Remark 7.3. One can consider tensor product of Fock modules with different twists F σ 1 u 1 ⊗ · · · ⊗ F σ d u d . According to Proposition 7.11 algebra W(sl n i , n i ) ⊗ U (Heis) acts on this space. Obtained representation is 'irregular' (cf. [Nag15]). In Section D.2 we consider an intertwiner between irregular and (graded completion of) regular representation.

Explicit formula for bosonization
Below we will write explicit formula for bosonization of W q (sl nd , n d). This bosonization comes from action of W(sl nd , n d) on F Let us define representation F η ⊗C[Q d (n) ] (cf. Section 4.2). Lattice Q d (n) consist of elements λ i b Q i b such that λ i b ∈ Z and for any i it holds b λ i b = 0. Define an action First factor F η is a Fock space for subalgebra η i b [k] for k = 0. We can consider F η ⊗ C[Q d (n) ] as representation of whole Heisenberg algebra as follows: η i b [k] for k = 0 acts on first factor, η i a [0] acts on the second factor by (7.51). Note, that also C[Q d (n) ] acts on F η ⊗ C[Q d (n) ]. Let us introduce notation Proposition 7.13. There is an action of W q (sl nd , n d) on F η ⊗ C[Q d (n) ] given by formulas (we consider product over such r that a − 1 r b for a > b and b − 1 r a for b > a).

Explicit formulas for strange bosonization
To write formulas for strange bosonization we need to consider Heisenberg algebra generated by ξ i [k] for i = 1, . . . d and k ∈ Z. Relations are given by linear dependence and commutation relations for either n k 1 or n k 2 (7.58) Denote corresponding Fock module by F ξ .
Proposition 7.14. There is an action of W q (sl nd , n d) on F ξ given by Obtained representation is isomorphic to F Wq(sl nd ,n d) u 1 ,...,u d .
Proof. The proof is analogous to proof of Proposition 7.13. The only difference is that we have to use (4.20), (4.21) instead of (4.17), (4.18). This representation is isomorphic to F Wq(sl nd ,n d) u 1 ,...,u d since it also corresponds to F (n ,n) u 1 ⊗ · · · ⊗ F (n ,n) u d .

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 W q (sl n , n tw ). Denote d = gcd(n tw , n). · · · jt kt < − ntw n and 1 k i n.
Sketch of the proof. Let us consider operator is a basis of F H (this basis coincide with a basis of complete homogeneous polynomials up to renormalization of Heisenberg algebra generators). One the other hand, here lower terms are taken with respect to lexicographical order. Proof. Theorem 7.1 implies that the natural map U (Diff 0 q ) → U (Diff q )/J n,ntw is surjective. Hence Verma module is generated by non-commutative monomials in E i and H j applied to λ gl . Using gl (with the same condition on k i and j i ).
Note that if jt kt > − ntw n then such vector is 0; moreover if jt kt = − ntw n then E kt [j t ] acts by multiplication on a constant, hence it can be excluded. Also note that E k [j] ∈ J n,ntw for k n + 1, therefore we assume k i n. The Proposition 7.15 is proven.
Definition 7.5. Verma module V Wq(sln,ntw) λ 1 ,...,λ d is a module over W q (sl n , n tw ) with cyclic vector λ sl and relations T tw k [r] λ sl = 0 for r > 0 (7.72) Introduce grading on W q (sl n , n tw ) by deg T tw k [r] = −r. Verma module V Wq(gl n ,ntw) λ 1 ,...,λ d is a graded module with grading defined by deg λ sl = 0. To simplify notation for comparison of sl n and gl n cases, we will assume below that µ = 1 (cf. Remark 7.4).
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. 8 Restriction on Diff Λ q for general sublattice We generalize results of Section 5 for arbitrary sublattice. For applications in Section 9 we will need only case of sublattice Λ 0 = span(e 1 , ne 2 ) ⊂ Z 2 .

Decomposition of restriction
Let Λ ⊂ Z 2 be a sublattice of finite index. Let us choose basis w 1 , w 2 of lattice Λ such that w 1 = (r, n tw ) and w 2 = (0, n). Let d be the greatest common divisor of n and n tw .
Theorem 8.1. There is an isomorphism of Diff q modules Hence it is enough to consider case r = 1.
We will use realization of F constructed in Proposition 5.5. 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 α = 0, . . . , d − 1 let Q α,lα (n/d) be a subset of lattice P (n) consisting of elements a≡α mod dl a Q a such that a≡α mod dl a = l α .

Note that
.
Or equivalently Let F n d a,α be Fock module for Heisenberg algebra generated by a b [k] for b ≡ α mod d. Then Let us show that (8.2) is a decomposition of Diff q -modules (moreover, that it leads to decomposition (8.1)). Let us define here a,b = r (−1) ar[0] (product over such r that a − 1 r b for a > b and b − 1 r a for b > a). Sketch of the proof. Let us define˜ a,b = r (−1) ar[0] (product over r satisfying above inequalities and condition r ≡ α mod d. One can check that there exists an index set I such that conjugation of E tw α (z) and F tw α (z) by (i,j)∈I (−1) a i [0]a j [0] will turn a,b to˜ a,b . Theorem 4.2 finishes the proof.
On the other hand, formulas (5.27)-(5.29) implies Therefore embedding of vector space from first row of following commutative diagram leads to second row embedding of Diff q -modules Corollary 8.2. Following Diff q -modules are isomorphic Remark 8.1. Lattice Λ admits another basis v 1 = (N, 0), v 2 = (R, d). There is an isomorphism
Remark 8.2. Another way to proof Proposition 8.3 is to derive it from Proposition 7.12 (since isomorphism (8.1)). Beware inconsistency of our notation in (7.48) and (8.10). Let us rewrite (7.48) Let us consider subalgebra of Heisenberg algebra generated by J k = a k for n k. Denote corresponding Fock module by F J .
Corollary 8.4. There is an action of W q (sl n , n tw ) on F J given by Obtained representation corresponds to F .
Proof. Follows from Theorem 7.1 and Proposition 8.3 Remark 8.3. Let us consider non-twisted case n tw = 0. Then d = n and total sign −(−1) d n = 1. More accurately, the coefficient is a root of unity of degree n (cf. Remark 7.4). Nevertheless, this freedom disappears if we require T 1 (z) = n + o( ) for = log q (this is a standard setting for classical limit).
Example 8.1. Odd bosonization is a particular case of strange bosonization for n = 2 and n tw = 0.
Consider classical limit q → 1. It is convenient to assume q = e and → 0. If there exist an expansion then modes of current L(z) = L n z −n−2 form 'not q-deformed' Virasoro algebra. Note that where J odd (z) = 2 r J r z − r 2 −1 . Hence 9 Relations on conformal blocks 9.1 Whittaker vector In this section we define and study basic properties of Whittaker vector W (z|u 1 , . . . , u N ) ∈ F u 1 ⊗ · · · ⊗ F u N . We will restrict ourself to case when F u 1 ⊗ · · · ⊗ F u N is irreducible.
In papers [Neg15a] [Tsy17] Whittaker vector is defined geometrically. We will define Whittaker vector by algebraic properties (cf. [Neg15a,Prop. 4.15]). Then we will proof that these properties define Whittaker vector uniquely up to normalization if the module F u 1 ⊗ · · · ⊗ F u N is irreducible.
Remark 9.1. Whittaker vector is an element of graded completion of F u 1 ⊗· · ·⊗F u N . Abusing notation, we use the same symbols for modules and their completions.
Remark 9.2. 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.2 clarify origin of this eigenvalues.
Theorem 9.1. If F u 1 ⊗ · · · ⊗ F u N is irreducible, then there exist unique Whittaker vector.
One can find a proof of Theorem 9.1 in Appendix D. This statement can be considered as a part of folklore; unfortunately, we do not know a precise reference for the theorem. Note that conditions (9.9)-(9.11) and conditions (9.6)-(9.8) coincide. Hence each component of W (z 1/n |u) also satisfy those conditions, i.e. coincide with Whittaker vector up to normalization. 9.1.1 Whittaker vector for F u Recall that we use notation c(λ) = c∈λ c(s).
Proposition 9.3. We have an expansion of vector W (z|u) in the basis |λ To proof the Proposition we need the following lemmas.
Lemma 9.4. Following vectors in F u coincide To finish the proof the proof let us recall that there is an identification of space of symmetric polynomials and Fock module F a α given by s λ → |λ and p k → a −k (see [KR87]).
Lemma 9.5. Following vectors in F u coincide Proof. Recall that we have defined an operator I τ by (3.23). Let us calculate Proposition 3.8 implies that Using Cauchy identity for another specialization we see that RHS of (9.17) and (9.18) coincide.
Remark 9.5. Note that we did not use Theorem 9.1 in the proof of Proposition 9.3. Moreover, we have proven a particular case of the Theorem for W (z|u).

Whittaker vector and restriction on sublattice
Let us recall interpretation of decomposition (8.7) in terms of boson-fermion correspondence. One can We argue by construction that decomposition (8.7) correspond to (9.20).
(i) Hooks of λ are in bijection with tuples Proof. The n-fermion Fock space F nψ is isomorphic to tensor product F ψ ⊗· · ·⊗F ψ . The n-Heisenberg highest vectors are products |l 0 ⊗ · · · ⊗ |l n−1 . After identification of F nψ with one F ψ , these products becomes (3.16) for special λ. Such diagrams λ are called n-cores.
Combinatorially boson-fermion correspondence is a correspondence between Maya diagrams and charged partitions (λ, l), see e.g. [Neg15b, Section 6.4] or [FM17]. The boxes of the partition corresponds to the pairs of white and black points in Maya diagram such that coordinate of white point is greater then coordinate of black point. The hook length equals difference between coordinates of white and black points (cf. [Neg15b, Section 6.4]). This proves (i).
Lemma 9.7. Let l i > l j . Let us consider hooks with fixed i and j (see Proposition 9.6).
There are exactly l i − l j − k such hooks of length nk + i − j for all possible k.
Proof. There exist a unique pairing on Fock space such that a k is dual to −a −k . Since algebra Diff q is generated by modes of E(z) and F (z), it remains to check Shapovalov property for them. Formulas (3.2) and (3.
Remark 9.6. Note that this pairing differs from the pairing defined in Section 3.1. More precisely, in Section 3.1 we required a k to be dual to a −k , not −a −k .
Proposition 9.13. Shapovalov pairing on F Orthogonality with all other summands also follows from Proposition 9.12 and irreducibility.
Remark 9.7. Let us comment on another way to proof orthogonality mentioned in Proposition 9.13. All direct summands are pairwise non-isomorphic. Hence there is no non-zero pairing for all other pairs of direct summands.
i =j Proof. Let l i − l j > 0. It is straightforward to check that for i > j and v ij = q n+i−j n for i < j. Formulas (9.32)-(9.33) implies the following assertions. For i > j Using identities (9.34)-(9.35), we obtain i =j To finish the proof, it remains to clarify the sign. This product already appeared as the product over all hooks. For diagram λ the number of hooks is |λ|.
Multiplying RHS of (9.37) and (9.38) by z Note that here we applied Proposition 9.14.

A Regular product
In this section we develope general theory of regular product. Term 'regular product' should be considered as an opposite to regularized (i.e. normally ordered) product. Let A(z) = k∈Z A k z −k be a formal power series with coefficients in End(V ) for a vector space V .
Let G(z, w) = k,l∈Z G k,l z −k w −l be a formal power series in two variables with operator coefficients. The operators G k,l acts on a vector space V .
Definition A.2. We will call G(z, w) regular if for any N and for any v ∈ V there are only finitely many G k,l such that k + l = N and G k,l v = 0.
If a current G(z, w) in two variables is regular one can substitute w = az and obtain well-defined power series G(z, az) for any a ∈ C.
Let A(z) and B(w) be two smooth formal power series with operator coefficients. Recall definition of normal ordering. Denote A + (z) = k 0 A −k z k and A − (z) = k<0 A −k z k . The sign (−1) depends on parity of A(z) and B(z) in the standard way. Note that smooth formal power series in two variables : A(z)B(w) : is regular. Formal power series A(z) and B(z) are called local (in weaker sence) if where s 1 , . . . , s N ∈ Z ≥0 , a 1 , . . . a N ∈ C and C Proposition A.1. If currents A(z) and B(w) are smooth and satisfy (A.1), then the following product (a 1 z − w) s 1 · · · (a N z − w) s N A(z)B(w) is regular.
Example A.2. Let A(z) = B(z) = E(z). Then Therefore, Let us comment on deep meaning of formula (A.9). One can present algebra Diff q using currents E k (z) (currents of Lie algebra type) or E k (z) (currents of q-W algebra type). This two series of currents are connected in non-trivial way starting from k = 2. For k = 2 they are related by (A.9). For general k see formula (7.17) in [Neg18]. (of course we assume that these regular products are well defined).
Proof. To study E(z)E(w) k we will consider OPE E(z)E(w 1 ) · · · E(w k ) and substitute w i = w. Only term (z − q w i ) −1 (z − q w j ) −1 . . . can give poles of order higher than 1 after substitution. OPE is symmetric on z, w 1 , . . . , w k as a rational function. We will consider order E(w i )E(z)E(w j ) · · · . According to Proposition B.4, the term (z − q w i ) −1 (z − q w j ) −1 . . . does not appear.

C Homomorphism from Diff q to W -algebra
This appendix is devoted to proof of Propositions 7.8. The proof is a straightforward check of relation from Proposition 2.2. The relations will be checked for operators j z −j , c = n, c = n tw , (C.1) Proposition C.1. Relations (2.8), (2.9) are satisfied.
Let us denote W q (gl n ) := W q (gl n , 0) = U (Diff q )/J n,0 . Denote Verma module for W q (gl n ) by V Wq(gl n ) λ 1 ,...,λ d (cf. Definition 7.4). Definition D.1. For each graded Diff q module M let us define Shapovalov dual module M ∨ . As a vector space M ∨ is graded dual to M . Action is defined by requirement that canonical pairing M ∨ ⊗ M → C is Shapovalov.
Finally note that involution E a,b → E −a,−b maps ideal J n,0 to J n,0 (maybe with different µ). Hence if M is a W q (gl n )-module then so is M ∨ .
Proposition D.2. Let u i /u j = q k for any k ∈ Z (cf. Lemma 9.1). There is no more than one Whittaker vector W (z|u 1 , . . . , u N ) ∈ F u 1 ⊗ · · · ⊗ F u N .
Proof. Denote by n a subalgebra of W q (gl n ) generated by E k [d] and H j for k = 1, . . . , n − 1, d > 0 and j > 0. Analogously, let n ∨ be a subalgebra of W q (gl n ) generated by F k [−d] and H −j for k = 1, . . . , n−1, d > 0 and j > 0. Note that involution E a,b → E −a,−b induces an involoution on W q (gl n ) which swaps n and n ∨ .
Consider F u 1 ⊗ · · · ⊗ F un as a W q (gl n )-module. Whittaker vector is an eigenvector for n. Hence, it is enough to show that F u 1 ⊗· · ·⊗F un is cocyclic for n. Equivalently, we need to proof that Shapovalov dual module (F u 1 ⊗ · · · ⊗ F un ) ∨ ∼ = F q/un ⊗ · · · ⊗ F q/u 1 is cyclic for n ∨ .

D.2 Construction of Whittaker vector
Let (n 1 , n 1 ) and (n 2 , n 2 ) be a basis of Z 2 . Remark D.1. Actually operators Φ and Φ * maps to graded completion of F (n 1 +n 2 ,n 1 +n 2 ) −q −1/2 uv and F (n 1 ,n 1 ) u ⊗ F (n 2 ,n 2 ) v correspondingly. Abusing notation, we will use the same symbol for module and its completion. Moreover, we are going to consider composition of such Φ * ; there appear an infinite sums as a result of such composition (a priori this sum does not make sense). We will use calculus approach to infinite sums; below we will provide sufficient condition for convergence of the series.
Proof. This is equivalent to Proposition 9.9.
Remark D.2. Recall that existence of Whittaker vector can be seen from geometric construction (see [Neg15a] and [Tsy17]).
Proof of Theorem 9.1. Existence and uniqueness follows from Theorem D.2 and Proposition D.2 correspondingly.
One can find notion of Whittaker vector for W q (sl n ) in the literature (see [Tak10]). In this section we will explain connection between notion of Whittaker vector W sln m (z|u 1 , . . . , u n ) and Whittaker vector W (z|u 1 , . . . , u n ) for Diff q (see Definition 9.1). Our plan to explain this connection is as follows.
First we define Whittaker vector with respect to W q (gl n ) (we denote it by W gl n m (z|u 1 , . . . , u n )). Then we will see, that on the one hand, the vector W gl n m (z|u 1 , . . . , u n ) is connected with W sln m (z|u 1 , . . . , u n ); on the other hand it is connected with W (z|u 1 , . . . , u n ).
Corollary D.10. There exists unique W m (z|u 1 , . . . , u n ) if u i /u j = q k .
Proof. We already know uniqueness of W gl n m (z|u 1 , . . . , u n ) and existence of W m (z|u 1 , . . . , u n ) from Proposition D.8 and Corollary D.10 correspondingly. So it is sufficient to show that W m (z|u 1 , . . . , u n ) satisfies properties of W gl n m (z|u 1 , . . . , u n ). Last assertion follows from formula (D.1) (also see (D.17)).
Proof. Follows from Lemma D.7 and Proposition D.11