Elliptic Stable Envelopes and Finite-dimensional Representations of Elliptic Quantum Group

We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we show a correspondence of the Gelfand-Tsetlin bases (resp. the standard bases) to the fixed point classes (resp. the stable classes) in the equivariant elliptic cohomology $E_T(X)$ of the cotangent bundle $X$ of the partial flag variety. As a result we obtain a geometric representation of the elliptic quantum group on $E_T(X)$.


Introduction
It has long been conjectured that there is a parallelism between the infinite dimensional (quantum) algebras and (equivariant) cohomology, K-theory, and elliptic cohomology [17,19]. In [16,39,40], finite-dimensional representations of symmetrizable Kac-Moody algebras g were constructed in terms of homology groups of quiver varieties. Their extension to the quantized universal enveloping algebras U q (g) and to their affinization were constructed on equivariant K-theory of quiver varieties [18, 20, 39-41, 53, 55]. Note that Yangian Y (g) is obtained by replacing equivariant K-theory by equivariant homology [41,54]. The basic tool in these works are convolution operation and correspondences in homology and equivariant K-theory. See for example [4]. However the elliptic case still remains conjecture.
Stable envelopes introduced by Maulik and Okoukov [35] are new tools to tackle this problem.
For a quiver variety X, stable envelope is a map from the equivariant cohomology of the torus A-fixed point set X A to the equivariant cohomology of X. It was extended to equivariant Ktheory [43] and equivariant elliptic cohomology [1]. In terms of stable envelopes Maulik and Okoukov constructed rational R matrices geometrically and obtained a geometric realization of the Yangian Y Q associated with a quiver Q [35]. Such geometric construction of R matrices was extended to the trigonometric [43] and the elliptic [1] cases. The stable envelopes also proved to be useful in solving integrable systems [2,35,49].
This new approach was enhanced by a discovery of a connection to the weight functions appearing in the hypergeometric integral solutions to the difference KZ equations. Gorbounov, Rimányi, Tarasov and Varchenko found an identification of rational weight functions with stable envelopes for torus-equivariant cohomology of the partial flag variety T * F λ [21] and extended this to the trigonometric ones for the equivariant K-theory [44]. Furthermore they succeeded to construct a geometric representation of the Yangian Y (gl N ) [21] and the quantum affine algebra U q ( gl N ) [44] on the equivariant cohomology and the equivariant K-theory, respectively.
In these works, a correspondence between finite-dimensional representations of quantum groups on the Gelfand-Tsetlin basis of the tensor product of the vector representations and geometric representations is a key to construction. Furthermore Felder, Rimányi and Varchenko [14] proposed a geometric representation of the dynamical elliptic quantum group E τ,y (gl 2 ) by using the sl 2 type elliptic weight function obtained in [13,50].
The elliptic weight functions of type sl N were derived in the previous paper [33] by using representation theory of the elliptic quantum group U q,p ( sl N ) [7,24,28,29]. The U q,p ( sl N ) is a Drinfeld realization of the dynamical elliptic quantum group and is isomorphic to the central extension of Felder's elliptic quantum group E q,p ( sl N ) [32]. Furthermore, in [33] their properties such as triangularity, transition property, orthogonality, quasi-periodicity and shuffle algebra structure were investigated. Comparing these properties with those of the elliptic stable envelopes in [1], we conjectured that the elliptic weight functions can be identified with the elliptic stable envelopes. Some of similar but slightly different results were presented in [46].
The purpose of this paper is to formulate a geometric representation of the higher rank dynamical elliptic quantum group associated with sl N . Constructing the Gelfand-Tsetlin basis of the tensor product of the n-vector representations explicitly (Theorem 4.5), we obtain finite-dimensional representations of E q,p ( gl N ) on it. In particular, we obtain an action of the half-currents of E q,p ( gl N ) and of the associated elliptic currents of U q,p ( sl N ) on the Gelfand-Tsetlin basis (Theorem 4.7 and Corollary 4.8). The resultant representations are described in a combinatorial way by using the partitions of [1, n]. It turns out that in the trigonometric and non-dynamical limit their combinatorial structures coincide with those of U q ( sl N ) on the equivariant K-theory obtained by Ginzburg and Vasserot [20,55] and by Nakajima [41].
We then lift these representations to the geometric ones by identifying the elliptic weight functions with the elliptic stable envelopes. We make a direct comparison of the elliptic weight functions with the abelianization formula of the elliptic stable envelopes, which was obtained by Shenfeld [48] in the rational case and extended to the elliptic case in [1]. We also obtain an identification of certain specializations of the elliptic weight functions with the elliptic stable envelops restricted to the torus fixed points. In this restriction, the stable envelopes play a role of the change of basis matrix elements from the stable classes to the fixed point classes in E T (T * F λ ). This allows us to define the fixed point classes in E T (T * F λ ) as transformations from the stable classes.
We then find that this defining relation of the fixed point classes (5.15) is identical to the change of basis relation from the standard basis to the Gelfand-Tsetlin basis (4.2). Then a correspondence between the Gelfand-Tsetlin bases ( resp. the standard bases) and the fixed point classes (resp. the stable classes) in E T (T * F λ ) yields a definition of the actions of the half-currents of E q,p ( gl N ) and of the elliptic currents of U q,p ( sl N ) on the fixed point classes in E T (T * F λ ), and provides a geometric representation of the elliptic quantum group on E T (T * F λ ) (Theorem 5.1 and Corollary 5.2).
In [46], a similar formula for elliptic weight functions of type sl N and their triangularity and the orthogonality properties are presented without derivation. There the triangular property agrees with ours but the orthogonality property seems wrong due to a lack of the dynamical shift. There are also no formulas for the shuffle algebra in [46]. In addition, it seems that in [46] a different formulation of elliptic stable envelopes from the one in [1] is presented. The relation between them is not clear for us. However, we would like to stress that in [14,46] there are neither statements on the definition of the fixed point classes in E T (T * F λ ) nor the correspondence between the Gelfand-Tsetlin bases and the fixed point classes, which are the keys to our results. This paper is organized as follows. In Section 2 we prepare some notations including the elliptic dynamical R matrices. We also provide defining relations of the elliptic quantum groups E q,p ( gl N ) and U q,p ( gl N ), and their basic properties. Definition of the half-currents of E q,p ( gl N ) and thier relationship to the elliptic currents of U q,p ( gl N ) are also exposed. Section 3 is devoted to a summary of the properties of the elliptic weight functions obtained in [33], such as the triangular property, transition property, orthogonality, quasi-periodicity and the shuffle algebra structures. In Section 4, we discuss a construction of finite-dimensional representations of E q,p ( gl N ) and U q,p ( sl N ) on the Gelfand-Tsetlin basis of the tensor product of the vector representations. In particular we show that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight functions. In Section 5, we discuss an identification between the elliptic weight functions and the elliptic stable envelopes and give a geometric representation of E q,p ( gl N ) and U q,p ( sl N ) on E T (T * F λ ). In Appendix A we summarize a co-algebra structure of E q,p ( gl N ) and U q,p ( gl N ). In Appendix B we present a proof of our main Theorem 4.7. In Appendix C we present a direct check of Corollary 4.8 for the relation (2.33).

Preliminaries
Through this paper we follow the notations in [33]. We here list the basic ones.

The commutative algebra H
We regardh ⊕h * as the Heisenberg algebra by Similarly, • {P α , Q β } (α, β ∈h * ) : the Heisenberg algebra defined by Then we define CPǭ j + Cc, where (P + h)ǭ j is an abbreviation of Pǭ j + hǭ j .
• F = M H * : the field of meromorphic functions on H * .

q-integers, infinite products and theta functions
Let q be generic complex numbers satisfying |q| < 1.
Let r be a generic positive real number and set p = q 2r . We use the following Jacobi's odd theta functions.

2.3
The elliptic dynamical R-matrix of the sl N type N −1 face type Boltzmann weight [22] and can be obtained [30] by taking the vector representation of the universal elliptic dynamical R matrix [23].
The R ± (z, q 2s ) satisfies the dynamical Yang-Baxter equation where q 2h (l) j,k acts on the l-th tensor space V by q 2h (l) j,k v µ = q 2<ǭµ,h j,k > v µ , and the unitarity The elliptic quantum groups E q,p ( gl N ) and U q,p ( gl N ) We consider the dynamical elliptic quantum group realized in the two ways E q,p ( gl N ) and U q,p ( gl N ). The elliptic algebra E q,p ( gl N ) is a central extension of Felder's elliptic quantum group [11,32], whereas U q,p ( gl N ) is an elliptic and dynamical analogue [29,32] of Drinfeld's new realization of the quantum affine algebra U q ( gl N ) [5]. For the details of the definitions we refer the reader to Sec.3 and Appendix D.1 in [32] and Appendix A in [33].
For simplicity of presentation, we treat the elliptic algebra E q,p ( gl N ) as a unital associative algebra over F generated by ( the Laurent coefficients of) L + ij (z) (1 ≤ i, j ≤ N ) and the central element q ±c/2 . Let L + (z) = 1≤i,j≤N E ij L + ij (z). In the level k ∈ R representation, where c = k, the defining relations are given as follows.
One can express the half-currents in terms of the quantum minor determinant of the L operators [32]. For 1 ≤ a, b ≤ N , let us define L + (z) a,a = ( L + i,j (z)) a≤i,j≤N and Then we obtain Then the q-determinant of L + (z) is given by and belongs to the center of E q,p ( gl N ).
Moreover from Proposition 6.4 in [32], we have Corollary 2.4. For l = 1, · · · , N , the q-principal minor determinant q-det L + (z) ll is given by and belongs to the center of the subalgebra E q,p ( gl N −l+1 ) of E q,p ( gl N ).
We define the elliptic algebra E q,p ( sl N ) as the quotient algebra E q,p ( gl N )/ < K(z) − 1 >.

The elliptic algebra
For simplicity of presentation, we treat the elliptic algebra U q,p ( gl N ) as a unital associative algebra over F generated by (the Laurent coefficients of) the elliptic currents E j (z), F j (z), K + l (z) (1 ≤ j ≤ N − 1, 1 ≤ l ≤ N ). In the level-k (k ∈ R) representation, the defining relations are given in the sense of analytic continuation as follows. For g(P ), g(P + h) ∈ F, (2.20) Proposition 2.5. [32] The following product belongs to the center of U q,p ( gl N ).
We define the elliptic algebra U q,p ( sl N ) as the quotient algebra U q,p ( gl N )/ < K(z) − 1 >.

Furthermore let us set
to be a unital subalgebra generated by ( the Laurent coefficients of )

Dynamical L operators and half-currents
For later convenience ( see Sec.4) we introduce the dynamical L operators [24,28] by where π(h ǫ j ) = E jj . Then L ± (z, P ) commutes with the elements in F and satisfy the full Accordingly we define the dynamical half-currents as the corresponding Gauss coordinates of L ± (z, P ). Then the relation between the half-currents from L ± (z) and those from L ± (z, P ) is given as follows.

Elliptic Weight Functions
In this section we summarize some basic properties of the elliptic weight functions obtained in [33].

The elliptic weight functions of type sl N
We consider the following elliptic weight functions [33].
where for s ∈ [1, n] we defines by i µs,s = s and m µs,l+1 (s) by

Entire function version
Let us set The following gives an entire function version of the elliptic weight function.
Furthermore, in order to compare with the stable envelopes, it is convenient to consider the following expression. See Sec.5.3.
Remark. In the trigonometric (p → 0) and non-dynamical (neglecting the factors depending on P + h ) limit W I and W I coincide with W I and W I discussed in [44], respectively. See also [36,37].

Transition property
Note that since H λ (t, z) is a symmetric function in z 1 , · · · , z n , W I (t, z, Π) has the same property.

Orthogonality
Noting (3.6) and the remark in Sec.5.3 in [33], where E λ (t, z) is the same as E λ (t) in (3.8), we have the following property.
where n j=1ǭ µ j is the weight associated with λ (Sec.3.1), and In Sec.5.4, a consistency between this property and the formula in Theorem 4.5 becomes a key to obtain a geometric representation of the elliptic quantum group. 1

Quasi-periodicity
Remember that we set t a + rτ . From (2.4) and Proposition 3.1 we obtain the following statement.
Proposition 3.5. For I ∈ I λ , the weight functions W I (t, z, Π) has the following quasi-periodicity.
From Proposition 3.5 one can deduce a symmetric integral M × M matrix N and a vector ξ ∈ (C/rZ) M , which imply the following quadratic form Then by Appel-Humbert theorem [34], a pair (N, ξ) characterizes a line bundle L(N, ξ) : and cocycle
(2) F (t; z, Π) has the quasi-periodicity Let us consider the subspace space M Consider a graded C-vector space

11)
where In the LHS of (3.11), we set t (l) This endows M(z, Π) with a structure of an associative unital algebra with the unit 1.
In [14], a sl 2 version of the ⋆-product is given.
Let us consider the subspace of M(z, Π).
All the elements in M + (z, Π) satisfy the following pole and wheel conditions. For

Finite Dimensional Representations
In this section we construct finite dimensional tensor product representations of the elliptic quantum group E q,p ( gl N ) and U q,p ( gl N ) on the Gelfand-Tsetlin basis.

Finite dimensional tensor product representations
The level-0 action of the L-operator L ± (z) or the dynamical L-operator L ± (z, P ) introduced in Sec.2.4.4 is given by where Π * j,l = q 2P j,l as before. The action on the tensor product space is obtained by the co-algebra structure presented in Appendix A.
Proof. It is enough to show the n = 2 case.
To obtain the third equality we used (A.2).
It is also useful to write down the comultiplication fomula of the dynamical L-operator, which is equivalent to Proposition 4.1.
⊗ denotes the usual tensor product, by We consider the the Gelfand-Tsetlin (GT) basis in V z 1 ⊗ · · · ⊗ V zn . Following [44] we construct it as follows. Firstly we realize S n in terms of the elliptic dynamical R matrix in (2.6).

The Gelfand-Tsetlin basis
Let define S i (P ) by Then by using the dynamical Yang-Baxter equation (2.10) and the unitarity relation (2.11) one can show the following.
For λ ∈ N N , |λ| = n, I = I µ 1 ···µn ∈ I λ , we set We define the Gelfand-Tsetlin basis {ξ I } I∈I λ by where Let us consider the change of basis matrix X = (X IJ (z, P )) I,J∈I λ : Here we put the matrix elements in the decreasing order I max · · · I min . Then by construction, X is a lower triangular matrix. Furthermore the following remarkable relationship between X and the specialized elliptic weight functions becomes a key to obtain a geometric interpretation of the results in the next subsection. See Sec.5.4.
Proof. Let J = I µ 1 ···µ i µ i+1 ···µn ∈ I λ . By definition, Hence we obtain for µ i = µ i+1 , and for µ i > µ i+1 . Here we set Note that (4.4) and (4.5) determine the whole matrix elements in X recursively starting from On the other hand, from Proposition 3.3 with replacing Π by Π * we have Using we obtain in particular for µ i > µ i+1 Specializing t = s i (z) I and noting etc., we obtain from (4.7) and (4.8) if µ i = µ i+1 , and if µ i > µ i+1 . Therefore one finds that W J (z −1 I , z −1 , Π * q 2 n j=1 <ǭµ j ,h> ) satisfy the same recursion relations as (4.4) and (4.5) for X IJ (z, P ). In addition their initial conditions coincide: Example. The case N = 2, n = 3, λ = (2, 1). We have I λ = {I 211 I 121 I 112 } and where u i,j = u i − u j . On the other hand we have

Action of the elliptic currents
In order to derive an action of the elliptic quantum group on the GT basis {ξ I }, the following property of the symmetrization operators S i (P ) is useful [44].
Thanks to this proposition it suffices to construct an action of ∆ ′ (n−1) ( L ± (w)) on ξ I max .
• E + 3,2 (1/w, P ): similarly we have Note that I 22211 is the maximal partition in I (2,3,0) . We also obtain • F + 2,3 (1/w, P ): similaly we obtain Again note that I 33211 is the maximal partition in I (2,1,2) . From (4.1) we have Substituting this and using the identity we obtain • K + 2 (1/w): let us consider the action On the other hand we have The last equality follows from the identity .
Furthermore noting the formula (4.14) and using Propositions 2.6 and 2.11 we obtain the following level-0 action of the elliptic currents of U q,p ( sl N ) on the GT basis.
In the trigonometric and non-dynamical limit, the combinatorial structures of the formulas in this Corollary are the same as those in the geometric representation of U q ( sl N ) on the equivariant K-theory of the quiver variety of type A N −1 obtained by Ginzburg and Vasserot [20,55], and by Nakajima [41]. One can directly check that these actions of the elliptic currents satisfy the defining relations of the level-0 U q,p ( sl N ) in the same way as in [41,55]. We give a check of the most non-trivial relation (2.33) in Appendix C. Proof. The statement follows from a similar argument to Theorem 4.11 in [31] and E j (1/w)ξ 11···1 = 0 (j = 1, · · · , N − 1),

Equivariant elliptic cohomology Ell T (X)
For λ = (λ 1 , · · · , λ N ) ∈ N N , |λ| = n, let F λ denote the partial flag variety as before and consider the cotangent bundle X = T * F λ . Let us set T = A × C * , A = (C * ) n . The torus A has a natural action on F λ and the extra C * acts on the fibers of T * F λ → F λ by multiplication with weight . Let E = C * /p Z (|p| < 1). We regard the elliptic curve E as a group scheme over C. We follow [1,14,15,17,19,46] for the definition of the T -equivariant elliptic cohomology Ell T (X).
The basic facts on Ell T (X) are summarized as follows.
(1) The T -equivariant elliptic cohomology, Ell T (X), is a functor from finite T -spaces X to superschemes, covariant in both T and X, satisfying a set of axioms ( [19], 4.1 in [15] and 2.1.2 in [1] ). In particular, Ell T (X) is a scheme over Ell T (pt) ∼ = E n × E. Moreover associated with a construction of X as a hyper-Kähler quotient we have a collection of tautological vector bundles {C λ (l) } of rk = λ (l) (l = 1, · · · , N − 1) over X and a map where E (m) = E m /S m denotes the symmetric product of E. This map is expected to be an embedding near the origin of Ell T (pt) (2.2 in [1]).
(2) The Thom class map Θ : K T (X) → Pic(Ell T (X)) is a map of a T -equivariant complex vector bundle ξ to a line bundle L ξ T over Ell T (X). The line bundle L ξ T is called the Thom sheaf of ξ. See 2.3.2 in [1] and Definition 6.1 in [15]. in [1]). If f is proper, pushforward is a morphism [17] and (11) in [1].
(4) The dynamical parameter dependence is introduced by extending Ell T (X) to as a scheme over B T,X = Ell T (pt) × E Pic T (X) . The variables in the two factors of B T,X ,

Chamber structure
Let Hom grp (C * , A) be the space of one parameter subgroups ρ in A and Hom grp (C * , A) ⊗ Z R ⊂ Lie A be its real form. The latter space can be decomposed into finitely many chambers C defined as a connected component of the compliment of the union of hyperplanes given by ρ such that X ρ(C * ) = X A [42].
The A-fixed points on X are described by the partitions in I λ . Let X A be the A-fixed point locus in X and X A = I∈I λ F I a decomposition to connected components. Let ρ ∈ C. For every S ⊂ X A we define its attracting set and denote by Attr f (S) the full attracting set, which is the minimal closed subset of X that contains the diagonal S × S and is closed under taking Attr(·). We then define a partial ordering on {F I } by
Let ι : X A → X be the inclusion map, which is proper. For each chamber C of Lie A, one can consider the polarization T 1/2 X ∈ K T (X) of X and its restriction T 1/2 X| X A to X A . Let us denote by ind := T 1/2 X| X A ,>0 the attracting part of T 1/2 X| X A . We have and a translation For the line bundle U Ell T (X A ) on Ell T (X A ) × E Pic T (X) we set where ι * is the pull-back of line bundles from X to X A .
The elliptic stable envelop Stab C is defined to be a map of O B T,X -modules where Θ(T 1/2 X) denotes the Thom sheaf of a polarization, and · · · stands for a certain line bundle pulled back from Stab C is subjected to the following two conditions (3.3.4 in [1]).
(i) (triangularity) Let s K be an elliptic cohomology class supported on F K locally over B T,X .
Then Stab C (s K ) is supported on Attr f (F K ). In particular if F K < F I we have (ii) (normalization) Near the diagonal in X × F K , we have are the natural projection and inclusion maps.

Direct comparison with the elliptic weight functions
In this and the following subsections, we consider the elliptic weight functions W I (t, z, Π * ) obtained from W I (t, z, Π) in (3.6) by replacing Π j,l = q 2(P +h) j,l by Π * j,l = q 2P j,l , which also satisfy all the properties in Sec. 3 under the same replacement.
The symmetry structure in the target of (5.1) coincides with the one of the elliptic weight function W I (t, z, Π * ) with respect to the variables t (l) a (l = 1, · · · , N − 1, a = 1, · · · , λ (l) ). This suggests that {t (l) a } can be identified with the Chern roots of the tautological vector bundles over X [3]. This structure as well as the quasi periodicity in Proposition 3.5 allow us to identify the elliptic weight functions W I (t, z, Π * ) with meromorphic sections of line bundles over E T (X) near the origin of B T,X .
Let G be a reductive group acting on a vector space M . It induces the hamiltonian action on T * M . Let µ G be the corresponding moment map. Let S ⊂ G be the maximal torus and let π S : (Lie G) * → (Lie S) * be a projection, and set µ S = π S • µ G . For the hyper-Käler quotient the associated abelian quotient is a hypertoric variety called the abelianization of X.
Here we defined for s ∈ I (l) , µ s ≤ l, a . The factor is a contribution from T * Gr(λ (l) , λ (l+1) ) with the Kähler parameters Π * a . Note that for N > 2, µ i (l) a in general takes a value in the range [1, l] corresponding to the embedding structure F I (1) ⊂ · · · ⊂ F ) and F I (l+1) . Note that the resultant C µs,l+1 (s) coincides with the one in Proposition 3.1.
Then again the abelianization formula [1,48] yields the following expression of Stab C (F I ) for .
From (5.10) we obtain This is an elliptic and dynamical analogue of the formula in Theorem 5.2.1 in [48].
By using the identification (5.11) and the equivalence of the specializations , whereJ = σ 0 (J), we obtain the following identification.
where in the second equality we used the identity

Geometric representation
Let X = T * F λ and fix a chamber C as above. By definition, the stable classes Stab C (F K ) (K ∈ I λ ) are triangular with respect to the fixed point classes {[I]} I∈I λ in E T (X). See 3.3.4 in [1]. Namely we have the following expansion formula Here we chose a normalization by R(z I ) given in Proposition 3.4 for later convenience. We regard this as the definition of the fixed point classes.
From (5.12), we have . (5.14) Note also that by the replacement z → z −1 and Π → Π * −1 one can rewrite Proposition 3.4 as Then using this and (5.14) one can invert (5.13) and obtain On the basis of this correspondence as well as Theorem 4.7 and Corollary 4.8, we obtain the following statement on the level-0 action of E q,p ( gl N ) and U q,p ( sl N ) on E T (X).
Theorem 5.1. Under the same notation as Theorem 4.7, let us define the action of the halfcurrents K ± j (1/w), E ± j+1,j (1/w, P ), F ± j,j+1 (1/w, P ) on the fixed point classes by Then this gives an irreducible finite-dimensional representation of E q,p ( gl N ) on E T (X).
Corollary 5.2. The level-0 action of U q,p ( sl N ) on E T (X) is given by Remark. Similar correspondences between the Gelfand-Tsetlin basis and the fixed point classes were studied in [8,9,26,38,51]. In [38] for the level-(0,1) representation of the quantum toroidal algebra of type A, the Gelfand-Tsetlin basis on the q-Fock space [52] was identified with the fixed point basis of the equivariant K-theory of corresponding cyclic quiver variety [53]. Affine Yangian analogue of this result was obtained in [26]. In [8,9,51], certain geometric actions of the universal enveloping algebra U (gl N ) on the Laumon spaces, of the affine Yangian of type A Let A denote E q,p ( gl N ) or U q,p ( gl N ). The A is bi-graded over H * by A α,β = x ∈ A q P +h xq −(P +h) = q <α,P +h> x, q P xq −P = q <β,P > x ∀P + h, P ∈ H , and possesses two moment maps µ l , µ r : F → A 0,0 defined by The µ l and µ r satisfy where T α = e α ∈ C[R Q ] denotes the automorphism of F T α f = e α f (P, P + h)e −α = f (P + < α, P >, P + h+ < α, P >).
Let A and B be two H-algebras. The tensor product A ⊗B is the H * -bigraded vector space where ⊗ F denotes the usual tensor product modulo the following relation.
The tensor product A ⊗B is again an H-algebra with the multiplication (b ⊗ a)(d ⊗ c) = bd ⊗ ac (a, c ∈ A, b, d ∈ B) and the moment maps We also consider the H-algebra of the shift operators [6] Then we have the H-algebra isomorphism The two H-algebras E q,p ( gl N ) and U q,p ( gl N ) are equipped with the common H-Hopf algebroid structure [32] defined by the two H-algebra homomorphisms, the co-unit ε : A → D and the (oposit) co-multiplication ∆ ′ : A → A ⊗A and the algebra antihomomorphism S :
(B.1): Noting <ǭ j+1 , h j,j+1 >= −1, from (4.11) and (4.12) we have Then the equality follows from the identitȳ a,b∈I j+1 a =b where we set P = P j,j+1 and u ak = u a − u k etc. The second equality follows from the identity Similarly the RHS of (B.3) yields a,b∈I j+1 a =b a,b∈I j+1 a =b Taking the difference between the LHS and the RHS, we obtain a,b∈I j+1 a =b where the equality follows from the identity and we set Since f (u a , u b ) = f (u b , u a ) and (I a ′ ) b ′ = (I b ′ ) a ′ (a, b ∈ I j+1 , a = b), the summation in (B.7) vanishes.
where z a = q 2ua , w = q 2v . Now let us check the relation (2.33).
2) the case i = j − 1: It is easy to show (I ′ a ) b ′ = (I b ′ ) ′ a , ∀a, b ∈ I j , (a = b). Then whereas F j (q j−N +1 /w 2 )E j−1 (q j−N /w 1 )ξ I = Ce Qα j−1 a,b∈I j a =b δ(z a /w 2 )δ(z b /w 1 ) Here we set (C. 2) The last equality follows from (2.38). Noting Hence we obtain Let us set the eigenvalue of H + j (q j−N +1 /w) on ξ I by ̺h j (v) with ̺ in (2.38) i.e.
Then we have for a ∈ I j and b ∈ I j+1 , Then from Lemma C.2 and (C.2), we obtain the desired formula.