Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology
Giovanni Felder Richárd Rimányi Alexander Varchenko












Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology

Elliptic Dynamical Quantum Groups
 and Equivariant Elliptic Cohomology[This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications. The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.htmlhttps://www.emis.de/journals/SIGMA/EHF2017.html]

Giovanni FELDER ^†, Richárd RIMÁNYI ^ and Alexander VARCHENKO ^

G. Felder, R. Rimányi and A. Varchenko

^† Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland
mailto:felder@math.ethz.chfelder@math.ethz.ch

^ Department of Mathematics, University of North Carolina at Chapel Hill,

 Chapel Hill, NC 27599-3250, USA
mailto:rimanyi@email.unc.edurimanyi@email.unc.edu, mailto:anv@email.unc.eduanv@email.unc.edu

Received April 30, 2018, in final form December 12, 2018; Published online December 21, 2018

We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based on weight functions and shuffle products. We construct an action of the dynamical elliptic quantum group associated with 𝔤𝔩_2 on the equivariant elliptic cohomology of the union of cotangent bundles of Grassmannians. The generators of the elliptic quantum groups act as difference operators on sections of admissible bundles, a notion introduced in this paper.

elliptic cohomology; elliptic quantum group; elliptic stable envelope

17B37; 55N34; 32C35; 55R40






§ INTRODUCTION

Maulik and Okounkov <cit.> have set up a program to realize representation theory of quantum groups of various kinds on torus equivariant (generalized) cohomology of Nakajima varieties. A central role is played by the stable envelopes, which are maps from the equivariant cohomology of the fixed point set of the torus action to the equivariant cohomology of the variety. Stable envelopes depend on the choice of a chamber (a connected component of the complement of an arrangement of real hyperplanes) and different chambers are related by R-matrices of the corresponding quantum groups. The basic example of a Nakajima variety is the cotangent bundle of the Grassmannian Gr(k,n) of k-planes in ℂ^n. The torus is T=U(1)^n× U(1), with U(1)^n acting by diagonal matrices on ℂ^n and U(1) acting by multiplication on the cotangent spaces. Then the Yangian Y(𝔤𝔩_2) acts on H_T(⊔_k=0^nT^*Gr(k,n)) and the action of generators is described geometrically by correspondences. It turns out that this representation is isomorphic to the tensor products of n evaluation vector representations with the equivariant parameters of U(1)^n as evaluation points and the equivariant parameter of U(1) as the deformation parameter of the quantum group. The choice of a chamber is the same as the choice of an ordering of the factors in the tensor product. The same holds for the affine quantum universal enveloping algebra U_q(𝔤𝔩_2) if we replace equivariant cohomology by equivariant K-theory. As was shown in <cit.>, the stable envelopes, which realize the isomorphisms, are given by the weight functions, which originally appeared in the theory of integral representations of solutions of the Knizhnik–Zamolodchikov equation, see <cit.>. Their special values form transition matrices from the tensor basis to a basis of eigenvectors for the Gelfand–Zetlin commutative subalgebra.

The recent preprint <cit.> of Aganagic and Okounkov suggests that the same picture should hold for equivariant elliptic cohomology and elliptic dynamical quantum groups and this is the subject of this paper. The authors of <cit.> define an elliptic version of the stable envelopes and show, in the example of the cotangent bundle of a projective space, stable envelopes corresponding to different orderings are related to the fundamental elliptic dynamical R-matrices of the elliptic dynamical quantum group E_τ,y(𝔤𝔩_2). Our paper is an attempt to understand the elliptic stable envelope in the case of cotangent bundles of Grassmannians. In particular we give a precise description of the space in which the stable envelope takes its values. Our construction of stable envelopes is based on elliptic weight functions. In Appendix <ref> we also give a geometric characterization, in terms of pull-backs to the cohomology of fixed points, in the spirit of <cit.>.

While our work is inspired by <cit.>, we do not know whether the two constructions are equivalent or not. The interesting project of understanding the exact relation between our construction and the construction of Aganagic–Okounkov requires more work.

Compared to equivariant cohomology and K-theory, two new features arise in the elliptic case. The first new feature is the occurrence of an additional variable, the dynamical parameter, in the elliptic quantum group. It also appears in <cit.>, under the name of Kähler parameter, in an extended version of the elliptic cohomology of Nakajima varieties. The second is a general feature of elliptic cohomology: while T-equivariant cohomology and K-theory are contravariant functors from T-spaces to supercommutative algebras, and can thus be thought of as covariant functors to affine superschemes,[The reader may safely ignore the super prefixes, as we only consider spaces with trivial odd cohomology, for which one has strictly commutative algebras.] in the elliptic case only the description as covariant functor to (typically non-affine) superschemes generalizes straightforwardly.

Our main result is a construction of an action of the elliptic quantum group associated with 𝔤𝔩_2 on the extended equivariant elliptic cohomology scheme Ê_T(X_n) of the union X_n=⊔_k=0^n X_k,n of cotangent bundles X_k,n=T^*Gr(k,n) of Grassmannians. The meaning of this is that we define a representation of the operator algebra of the quantum group by difference operators acting on sections of a class of line bundles on the extended elliptic cohomology scheme, which we call admissible bundles: up to a twist by a fixed line bundle, admissible bundles on Ê_T(X_k,n) are pull-backs of bundles on Ê_U(n)× U(1)(pt) (by functoriality there is a map corresponding to the map to a point and the inclusion of the Cartan subalgebra T→ U(n)× U(1)). The claim is that there is a representation of the elliptic quantum group by operators mapping sections of admissible bundles to sections of admissible bundles.

This paper may be considered as an elliptic version of the paper <cit.> where analogous constructions are developed for the rational dynamical quantum group E_y(𝔤𝔩_2).



 §.§ Notation
 For a positive integer n, we set [n]={1,…,n}. It K is a subset of [n] we denote by |K| its cardinality and by K̅ its complement. Throughout the paper we fix τ in the upper half plane and consider the complex elliptic curve E=ℂ/(ℤ+τℤ). The odd Jacobi theta function

    θ(z)=sinπ z/π∏_j=1^∞(1-q^j e^2π iz)(1-q^j e^-2π iz)/(1-q^j)^2,     q= e^2π iτ,

is normalized to have derivative 1 at 0. It is an entire odd function with simple zeros at ℤ+τℤ, obeying θ(z+1)=-θ(z) and

    θ(z+τ)=- e^-π iτ-2π izθ(z).




§ DYNAMICAL R-MATRICES AND ELLIPTIC QUANTUM GROUPS



 §.§ Dynamical Yang–Baxter equation

Let 𝔥 be a complex abelian Lie algebra and V an 𝔥-module with a weight decomposition V=⊕_μ∈𝔥^*V_μ and finite dimensional weight spaces V_μ. A dynamical R-matrix with values in End_𝔥(V⊗ V) is a meromorphic function (z,y,λ)↦ R(z,y,λ)∈End_𝔥(V⊗ V) of the spectral parameter z∈ℂ, the deformation parameter y∈ℂ and the dynamical parameter λ∈𝔥^*, obeying the dynamical Yang–Baxter equation

    R(z,y,λ-y h^(3))^(12)R(z+w,y,λ)^(13) R(w,y,λ-y h^(1))^(23)
        = R(w,y,λ)^(23)R(z+w,y,λ-y h^(2))^(13) R(z,y,λ-y h^(3))^(12)

in End(V⊗ V⊗ V) and the inversion relation

    R(z,y,λ)^(12)R(-z,y,λ)^(21)=Id

in End(V⊗ V). The superscripts indicate the factors in the tensor product on which the endomorphisms act non-trivially and h is the element in 𝔥^*⊗End(V) defined by the action of 𝔥: for example R(z,y,λ-y h^(3))^(12) acts as R(z,y,λ-yμ_3)⊗Id on V_μ_1⊗ V_μ_2⊗ V_μ_3.
[<cit.>] Let 𝔥≃ℂ^N be the Cartan subalgebra of diagonal matrices in 𝔤𝔩_N(ℂ). Let V=⊕_i=1^N V_ϵ_i the vector representation with weights ϵ_i(x)=x_i, x∈𝔥 and one-dimensional weight spaces. Let E_ij be the N× N matrix with entry 1 at (i,j) and 0 elsewhere. The elliptic dynamical R-matrix
 for 𝔤𝔩_N is[We use the convention of <cit.>. This R-matrix is obtained from the one introduced in <cit.> by substituting y=-2η and replacing z by -z.]
 
    R(z,y,λ) =∑_i=1^NE_ii⊗ E_ii+∑_i≠ jα(z,y,λ_i-λ_j) E_ii⊗ E_jj +∑_i≠ jβ(z,y,λ_i-λ_j) E_ij⊗ E_ji,

 where
 
    α(z,y,λ)= θ(z)θ(λ+y)/θ(z-y)θ(λ),     β(z,y,λ)= -θ(z+λ)θ(y)/θ(z-y)θ(λ).

 It is a deformation of the trivial R-matrix R(z,0,λ)=id_V⊗ V.

A dynamical R-matrix defines a representation of the symmetric group S_n on the space of meromorphic functions of (z_1,…,z_n,y,λ)∈ℂ^n×ℂ×𝔥^* with values in V^⊗ n. The transposition s_i=(i,i+1), i=1,…,n-1, acts as

    f↦ S_i(z,y,λ)s_i^*f,     S_i(z,y,λ)=R(z_i-z_i+1,y,λ-y∑_j=i+2^nh^(j))^(i,i+1) P^(i,i+1),

where P∈End(V⊗ V) is the flip u⊗ v↦ v⊗ u and s_i^* acts on functions by permutation of z_i with z_i+1.

To a dynamical R-matrix R there corresponds a category “of representations of the dynamical quantum group associated with R”. Fix y∈ℂ and let 𝕂 be the field of meromorphic functions of λ∈𝔥^* and for μ∈𝔥^* let τ_μ^*∈Aut(𝕂) be the automorphism τ_μ^*f(λ)=f(λ+yμ). An object of this category is a 𝕂-vector space W=⊕_μ∈𝔥^*W_μ, which is a semisimple module over 𝔥, with finite dimensional weight spaces W_μ, together with an endomorphisms L(w)∈End_𝔥(V⊗ W), depending on w∈ U⊂ℂ for some open dense set U, such that
=0pt

  (i) L(w)u⊗ fv=(id⊗τ^*_-μf) L(w)u⊗ v, f∈𝕂, u∈ V_μ,v∈ W.

  (ii) L obeys the RLL relations:
 
    R(w_1-w_2,y,λ-y h^(3))^(12)L(w_1)^(13) L(w_2)^(23)
        = L(w_2)^(23)L(w_1)^(13) R(w_1-w_2,y,λ)^(12).


Morphisms (W_1,L_W_1)→ (W_2,L_W_2) are 𝕂-linear maps φ W_1→ W_2 of 𝔥-modules, commuting with the action of the generators, in the sense that L_W_2(w) id_V⊗φ=id_V⊗φ  L_W_1(w) for all w in the domain of definition. The dynamical quantum group itself may be defined as generated by Laurent coefficients of matrix elements of L(w) subject to the RLL relations, see <cit.> for a recent approach in the case of elliptic dynamical quantum groups and for the relations with other definitions of elliptic quantum groups.

The basic example of a representation is the vector evaluation representation V(z) with evaluation point z∈ℂ. The vector representation has W=V⊗_ℂ𝕂 and

    L(w)v⊗ u=R(w-z,y,λ)v⊗τ_-μ^* u,     v∈ V_μ,     u∈ W.

Here τ^*_-μ (v⊗ f)=v⊗τ_-μ^*f for v∈ V and
f∈𝕂, and R acts as a multiplication operator.

More generally we have the tensor product of evaluation
representations V(z_1)⊗⋯⊗ V(z_n) with
W=V^⊗ n⊗𝕂, and, by numbering the factors of
V⊗ V^⊗ n by 0,1,…,n,

    L(w)v⊗ u =R(w-z_1,y,λ-y∑_i=2^nh^(i))^(01) R(w-z_2,y,λ-y ∑_i=3^nh^(i))^(02)⋯
    × R(w-z_n,y,λ)^(0,n) v⊗τ_-μ^* u,      v∈ V_μ,     u∈ W.

For generic z_1,…, z_n the tensor products does not essentially depend on the ordering of the factors: the operators S_i defined above are isomorphisms of representations

    V(z_1)⊗⋯⊗ V(z_i)⊗ V(z_i+1)⊗⋯⊗ V(z_n) → V(z_1)⊗⋯
        ⊗ V(z_i+1)⊗ V(z_i)⊗⋯⊗ V(z_n).

 It is convenient to consider L-operators L(w), such as (<ref>), which are meromorphic functions of w and are thus only defined for w in an open dense set. But one may prefer to consider
only representations with L(w) defined for all w∈ℂ. This may be obtained for the representation given by (<ref>) by replacing L(w) by the product of L(w) with ∏_a=1^nθ(w-z_a+y).



 §.§ Duality and gauge transformations

Suppose that R(z,y,λ) is a dynamical R-matrix with 𝔥-module V. Let V^∨=⊕_μ (V^∨)_μ with weight space (V^∨)_μ the dual space to V_μ. Then R^∨(z,y,λ)=(R(z,y,λ)^-1)^*, the dual map to R(z,y,λ)^-1, is a dynamical R-matrix with values in End_𝔥(V^∨⊗ V^∨). It is called the dual R-matrix to R.

Another way to get new R-matrices out of old is by a gauge transformation. Let ψ_V(λ) be a meromorphic function on ℂ×𝔥^* with values in Aut_𝔥(V). Let
ψ_V⊗ V(λ)=ψ_V(λ-y h^(2))^(1)ψ_V(λ)^(2)∈End_𝔥(V⊗ V). Then

    R_ψ(z,y,λ)=ψ_V⊗ V(λ)^-1R(z,y,λ)ψ_V⊗ V(λ)^(21)

is another dynamical R-matrix. The corresponding representations of the symmetric group are related by the isomorphism

    ψ_V^⊗ n(λ)=∏_i=1^nψ_V(λ-y∑_j=i+1^nh^(j))^(i).




 §.§ The elliptic dynamical quantum group E_τ,y(𝔤𝔩_2)


In this paper, we focus on the dynamical quantum group E_τ,y(𝔤𝔩_2). The corresponding R-matrix is the case N=2 of Example <ref>. With respect to the basis v_1⊗ v_1, v_1⊗ v_2, v_2⊗ v_1, v_2⊗ v_2,

    R(z,y,λ)= (
     
     1     0     0     0
    
     0    α(z,y,λ)    β(z,y,λ)     0
    
     0    β(z,y,-λ)    α(z,y,-λ)     0
    
     0     0     0    1
     ),

where λ=λ_1-λ_2. Since R depends only on the difference λ_1-λ_2 it is convenient to replace 𝔥 by the 1-dimensional subspace ℂ spanned by h=diag(1,-1). Then, under the identification 𝔥≅ℂ via the basis h, v_1 has weight 1 and v_2 has weight -1. Let (W,L) be a representation of E_τ,y(𝔤𝔩_2) and write L(w)=∑_i,j=1^2 E_ij⊗ L_ij(w). Then L_ij(w) maps W_μ to W_μ+2(i-j) and for f(λ)∈𝕂, L_i2(w)f(λ) =f(λ+y)L_i2(w) and L_i1(w)f(λ) =f(λ-y)L_i1(w).
[the vector representation V(z)] Let V=ℂ^2 with basis v_1, v_2, then
 
    L_11(w)v_1=v_1,     L_22(w)v_2=v_2,
    
     L_11(w)v_2= θ(w-z)θ(λ+y)/θ(w-z-y)θ(λ) v_2,    
     L_22(w)v_1= θ(w-z)θ(λ-y)/θ(w-z-y)θ(λ) v_2,
    
     L_12(w)v_1 =-θ(λ+w-z)θ(y)/θ(w-z-y)θ(λ) v_2,     
    L_21(w)v_2=-θ(λ-w+z)θ(y)/θ(w-z-y)θ(λ) v_1,

and the action on other basis vectors is 0.




 §.§ The Gelfand–Zetlin subalgebra

Let W be a representation of the elliptic dynamical quantum group E_τ,y(𝔤𝔩_2). Then L_22(w), w∈ℂ and the quantum determinant <cit.>

    Δ(w)=θ(λ-y h)/θ(λ) (L_11(w+y)L_22(w)-L_21(w+y)L_12(w))

generate a commutative subalgebra of End_𝔥(W). It is called the Gelfand–Zetlin subalgebra.



§ SHUFFLE PRODUCTS AND WEIGHT FUNCTIONS

Weight functions are special bases of spaces of sections of line bundles on symmetric powers of elliptic curves. They appear in the theory of hypergeometric integral representation of Knizhnik–Zamolodchikov equations. In <cit.> they were characterized as tensor product bases of a space of function for a suitable notion of tensor products. In this approach the R-matrices for highest weight representations of elliptic quantum groups arise as matrices relating bases obtained from taking different orderings of factors in the tensor product. We review and extend the construction of <cit.> in the special case of products of vector representations.



 §.§ Spaces of theta functions

 Let z∈ℂ^n,y∈ℂ,λ∈ℂ and define Θ^-_k(z,y,λ) to be the space of entire holomorphic functions f(t_1,…,t_k) of k variables such that
 =0pt
 
  * For all permutations σ∈ S_k, f(t_σ(1),…,t_σ(k))=f(t_1,…,t_k).
 
  * For all r,s∈ℤ, the meromorphic function
 
    g(t_1,…,t_k)=f(t_1,…,t_k)/∏_j=1^k∏_a=1^n θ(t_j-z_a)

 obeys
 
    g(t_1,…, t_i+r+sτ,…,t_k)= e^2π is(λ-ky)g(t_1,…,t_i,…,t_k).

 

 Let z∈ℂ^n, y∈ℂ, λ∈ℂ and define Θ^+_k(z,y,λ) to be the space of entire holomorphic functions f(t_1,…,t_k) of k variables such that
 =0pt
 
  * For all permutations σ∈ S_k, f(t_σ(1),…,t_σ(k))=f(t_1,…,t_k).
 
  * For all r,s∈ℤ, the meromorphic function
 
    g(t_1,…,t_k)=f(t_1,…,t_k)/∏_j=1^k∏_a=1^n θ(t_j-z_a+y),

 obeys
 
    g(t_1,…, t_i+r+sτ,…,t_k)= e^-2π is(λ-ky)g(t_1,…,t_i,…,t_k).

 

 These spaces are spaces of symmetric theta functions of degree n in k variables and have dimension n+k-1  k. Actually Θ^- depends on the parameters only through the combination ∑_a=1^n z_a+λ-ky and Θ^+ through the combination ∑_a=1^nz_a-λ-(n-k)y.

 For z∈ℂ and all k=0,1,2,…, Θ^-_k(z,y,λ) is a one-dimensional vector space
 spanned by
 
    ω^-_k(t;z,y,λ)=∏_j=1^kθ(λ-t_j+z-ky),

 Θ^+_k(z,y,λ) is a one-dimensional vector space spanned by
 
    ω^+_k(t;z,y,λ) =∏_j=1^kθ(λ+t_j-z+(1-k)y).


  For z∈ℂ^n, y,λ∈ℂ, Θ^-_k(z,y,λ)=Θ^+_k(z,y,-λ-(n-2k)y) and ω^+_k(t;z,y,λ)=(-1)^kω^-_k(t;z,y,-λ-(1-2k)y). It is however better to keep the two spaces distinct as they will be given a different structure.
 


 §.§ Shuffle products

Let Sym denote the map sending a function f(t_1,…,t_k) of k variables to the symmetric function ∑_σ∈ S_nf(t_σ(1),…,t_σ(k)).
 Let n=n'+n”, k=k'+k” be non-negative integers, z∈ℂ^n, z'=(z_1,…,z_n'), z”=(z_n'+1,…, z_n). Then the shuffle product
 
    * Θ^±_k'(z',y,λ+y(n”-2k”))⊗Θ^±_k”(z”,y,λ)→Θ^±_k(z,y,λ),

 sending f⊗ g to
 
    f*g(t)=1/k'!k”!Sym(f(t_1,…,t_k') g(t_k'+1,…,t_k)φ^±(t,z,y)),

 with
 
    φ^-(t,z,y)   =∏_j=1^k'∏_l=k'+1^k θ(t_l-t_j+y)/θ(t_l-t_j)∏_l=k'+1^k∏_a=1^n'θ(t_l-z_a) ∏_j=1^k'∏_b=n'+1^nθ(t_j-z_b+y), 
    φ^+(t,z,y)   =∏_j=1^k'∏_l=k'+1^k θ(t_j-t_l+y)/θ(t_j-t_l)∏_l=k'+1^k∏_a=1^n'θ(t_l-z_a+y) ∏_j=1^k'∏_b=n'+1^nθ(t_j-z_b),

 is well-defined and associative, in the sense that (f*g)*h=f*(g*h), whenever defined.

 In the formula for f*g in Proposition <ref> we can omit the factor 1/k'!k”! and replace the sum over permutations defining Sym by the sum over (k',k”)-shuffles, namely permutations σ∈ S_k such that σ(1)<…<σ(k') and σ(k'+1)<⋯<σ(k).

 This is essentially the first part of Proposition 3 of <cit.> in the special case of weights Λ_i=1. The proof is straightforward: the apparent poles at t_j=t_l are cancelled after the symmetrization since θ(t_j-t_l) is odd under interchange of t_j with t_l. Thus f*g is a symmetric entire function. One then checks that every term in the sum over permutations has the correct transformation property under lattice shifts.

 The maps * of Proposition <ref> define isomorphisms
 
    ⊕_k'=0^k Θ^±_k'(z',y,λ+y(n”-2k”)) ⊗Θ^±_k-k'(z”,y,λ) →Θ^±_k(z,y,λ)

 for generic z, y, λ.

We prove this Proposition in <ref> below.



 §.§ Vanishing condition

The shuffle product * preserves subspaces defined by a vanishing condition. It is the case of the fundamental weight of a condition introduced in <cit.> for general integral dominant weights.

Let (z,y,λ)∈ℂ^n×ℂ×ℂ. We define a subspace Θ̅^±_k(z,y,λ)⊂Θ^±_k(z,y,λ) by a vanishing condition:

    Θ̅^±_k(z,y,λ) = Θ^±_k(z,y,λ)   if k=0,1,
    {f  f(t_1,…,t_k-2,z_a,z_a-y)=0, 1≤ a≤ n,  t_i∈ℂ}   if k≥2.

 For n=1,
 
    Θ̅^±_k(z,y,λ) =Θ^±_k(z,y,λ)≅ℂ,   k=0,1,
    
     0,   k≥2.

 Indeed, the condition is vacuous if k≤ 1 and if k≥2 then ω^±_k(z;z-y,t_3,…)=θ(λ-ky)θ(λ+(1-k)y) times a nonzero function. For k=1, n≥1, Θ̅^±_1(z,y,λ) = Θ^±_1(z,y,λ). For k=2, n=2, Θ̅^±_2(z_1,z_2,y,λ) is a one-dimensional subspace of the three-dimensional space Θ^±_2(z,y,λ).

 The shuffle product restricts to a map

    ⊕_k'=0^k Θ̅^±_k' (z',y,λ+y(n”-2k”)) ⊗Θ̅^±_k-k' (z”,y,λ) →Θ̅^±_k(z,y,λ),

 which is an isomorphism for generic values of the parameters.

The proof is postponed to Section <ref> below. By iteration we obtain shuffle multiplication maps

    Φ̅_k^±(z,y,λ)⊕_Σ k_a=k⊗_a=1^n Θ̅^±_k_a(z_a,y,λ-y∑_b=a+1^n(2k_a-1)) →Θ̅^±_k(z_1,…,z_n,y,λ),

defined for (z,y,λ)∈ℂ^n×ℂ×ℂ and k=0,1,2,…. The direct sum is over the n  k n-tuples (k_1,…,k_n) with sum k and k_a∈{0,1}, a=1,…,n.
 The maps Φ̅_k^±(z,y,λ) are isomorphisms for generic (z,y,λ)∈ℂ^n×ℂ×ℂ.


Thus, for generic z,y,λ∈ℂ^n×ℂ×ℂ, Θ̅^±_k(z,y,λ) has dimension n  k and is zero if k>n.



 §.§ Duality

 The identification

    ϱ Θ^-_k(z,y,λ)→Θ^+_k(z,y,-λ-(n-2k)y)

of Remark <ref> (the identity map) restricts to an isomorphism

    Θ̅^-_k(z,y,λ)→Θ̅^+_k(z,y,-λ-(n-2k)y),

also denoted by ϱ. For f∈Θ^-_k' and g∈Θ^-_k” as in Proposition <ref>, the shuffle product ϱ(g)*ϱ(f) is well-defined and obeys

    ϱ(f*g)=ϱ(g)*ϱ(f).


 It is clear that the vanishing condition is preserved. The last claim follows from the identity
 
    φ^-(t,z,y)= φ^+(t_k'+1,…,t_k,t_1,…,t_k',z_n'+1,…,z_n,z_1,…,z_n',y)

 for the functions appearing in the definition of the shuffle product.


 For n=1 we have ϱ(ω^-_k)=(-1)^kω^+_k, see Example <ref>.



 §.§ Weight functions

For (z,y,λ)∈ℂ^n×ℂ×ℂ, let

    Θ̅^±(z,y,λ)=⊕_k=0^n Θ̅^±_k(z,y,λ).

It is an 𝔥-module with Θ̅^±_k of weight -n+2k. Let v_1, v_2 be the standard basis of ℂ^2. If n=1, we identify Θ̅^±(z,y,λ) with ℂ^2 via the map
ω^±_1↦ v_1, ω^±_0↦ v_2. Then Φ̅^±(z,y,λ)=⊕_kΦ̅^±_k(z,y,λ) is a linear map

    (ℂ^2)^⊗ n→Θ̅^±(z,y,λ).

It is a homomorphism of 𝔥-modules. Then a basis of (ℂ^2)^⊗ n is labeled by subsets I of [n]={1,…,n}: v_I=v_j(1)⊗⋯⊗ v_j(n) with j(a)=2 if a∈ I and j(a)=1 if a∈I̅, the complement of I.
 The weight functions ω^±_I(t;z,y,λ) are the functions
 
    ω^±_I(·;z,y,λ)=Φ̅^±(z,y,λ)v_I∈Θ̅^±(z,y,λ).


In particular, for n=1, ω^±_∅=ω^±_0, ω^±_{1}=ω_1^±. Corollary <ref> implies:
 Let (z,y,λ) be generic. The weight functions ω^±_I(·; z,y,λ) with I⊂ [n], |I|=k form a basis of the space Θ̅^±_k(z,y,λ) of theta
functions obeying the vanishing condition.

 For k=1 and n=1,2…, z∈ℂ^n, y∈ℂ, λ∈ℂ, a=1,…,n,
 
    ω^-_{a}(t;z,y,λ) =θ(λ-t+z_a+y(n-a-1)) ∏_b=1^a-1θ(t-z_b) ∏_b=a+1^n θ(t-z_b+y),
    ω^+_{a}(t;z,y,λ) =θ(λ+t-z_a+y(n-a)) ∏_b=1^a-1θ(t-z_b+y) ∏_b=a+1^n θ(t-z_b).




 §.§ R-matrices

Note that while Θ̅^±_k(z,y,λ) is independent of the ordering of z_1,…,z_n the map Φ̅_k^± does depend on it and different orderings are related by R-matrices, as we now describe. We
define R-matrices R^±(z,y,λ)∈End_𝔥(ℂ^2⊗ℂ^2) by

    R^±(z_1-z_2,y,λ) =Φ̅^±(z_1,z_2,y,λ)^-1Φ̅^±(z_2,z_1,y,λ)P,

where Pu⊗ v=v⊗ u is the flip of factors. Up to duality and gauge transformation, these R-matrices coincide with the elliptic R-matrix of Section <ref>:
  =0pt
 
  (i) Let s_i∈ S_n be the transposition (i,i+1). Then
 
    Φ̅^±(s_iz,y, λ) =Φ̅^±(z,y,λ) R^±(z_i-z_i+1,y,λ -y∑_j=i+2^nh^(j))^(i,i+1)P^(i,i+1).

 
  (ii) The R-matrices R^± obey the dynamical Yang–Baxter equation (<ref>) and the inversion relation (<ref>).
 
  (iii) With respect to the basis v_1⊗ v_1, v_1⊗ v_2, v_2⊗ v_1, v_2⊗ v_2 of ℂ^2⊗ℂ^2,
 
    R^-(z,y,λ)= (
     
     1     0     0     0
    
     0    α(-z,y,-λ)    β(-z,y,λ)     0
    
     0    β(-z,y,-λ)    α(-z,y,λ)     0
    
     0     0     0    1
     )=R^∨(z,y,λ)

 is the dual R-matrix, see Section <ref>, with the standard identification of ℂ^2 with (ℂ^2)^* and
 
    R^+(z,y,λ)= (
     
     1     0     0     0
    
     0    α(z,y,-λ)    β(z,y,λ)     0
    
     0    β(z,y,-λ)    α(z,y,λ)     0
    
     0     0     0    1
     )= R_ψ(z,y,λ)

 is the gauge transformed R-matrix with
 
    ψ(λ)=(
     θ(λ)θ(λ-y)     0 
    
     0     1
     ).

 


 Let (z,y,λ)∈ℂ^n×ℂ×ℂ be generic and set
 
    S_i(z,y,λ)= R(z_i-z_i+1,y,λ-y∑_j=i+2^nh^(j))^(i,i+1)P^(i,i+1)∈End_𝔥((ℂ^2)^⊗ n),

 i=1,…,n-1, cf. (<ref>).
 =0pt
 
  (i) For t∈ℂ^k, let ω^-(t;z,y,λ)=∑_I⊂[n], |I|=kω^-_I(t;z,y,λ)v_I. Then
 
    ω^-(t;z,y,λ)=S_i(z,y,λ) ω^-(t;s_iz,y,λ).

 
  (ii) Let ψ_V^⊗ n(λ)=∏_i=1^nψ(λ-y ∑_j>ih^(j))^(i), cf. Section <ref>. Then
 
    Φ̅^+(s_iz,y,λ)ψ_V^⊗ n(λ)^-1 =Φ̅^+(z,y,λ)ψ_V^⊗ n(λ)^-1 S_i(z,y,λ).

 




 §.§ A geometric representation

Let z_1,…,z_n,y,λ be generic and w∈ℂ. Recall that we identify Θ̅^+(w,y,λ) with V=ℂ^2 via the basis ω^+_1, ω^+_0. Consider the shuffle products[The compressed notation we are using might be confusing: the map p_+ is actually defined on ⊕_k Θ̅^+(w,y,λ+(n-2k)y)⊗Θ̅^+_k(z,y,λ). The identification of the first factor with V depends on k through the λ-dependence of the basis vectors ω^+_i.]

    p_+ V ⊗Θ̅^+(z_1,…,z_n,y,λ)→Θ̅^+(w,z_1,…,z_n,y,λ),
    
    p_- Θ̅^+(z_1,…,z_n,y,λ-yh^(2))⊗ V →Θ̅^+(w,z_1,…,z_n,y,λ).

Then varying w and denoting P the flip of tensor factors, we get a homomorphism

    ℓ(w,y,λ)=p_+^-1∘ p_-∘ P∈Hom(V⊗Θ̅^+(z,y,λ-y h^(1)),V⊗Θ̅^+(z,y,λ)).

By construction it obeys the dynamical Yang–Baxter equation

    R^+(w_1-w_2,y,λ-y h^(3))^(12)ℓ(w_1,y,λ)^(13)ℓ(w_2,y,λ-y h^(1))^(23)
         = ℓ(w_1,y,λ)^(23)ℓ(w_2,y,λ-y h^(2))^(13) R^+(w_1-w_2,y,λ-y h^(3))^(12)

in Hom(V⊗ V⊗Θ̅^+(z,y,λ-y(h^(1)+h^(2))), V⊗ V⊗Θ̅^+(z,y,λ)). By varying λ we obtain a representation of the elliptic dynamical quantum group as follows. Let (z,y)∈ℂ^n×ℂ be generic and consider the space Θ̅^+_k(z,y)_reg of holomorphic functions f(t,λ) on ℂ^k×ℂ such that for each fixed λ, t↦ f(t,λ) belongs to Θ̅^+(z,y,λ). It is a module over the ring 𝒪(ℂ) of holomorphic functions of λ. We set

    Θ̅^+_k(z,y)=Θ̅^+_k(z,y)_reg⊗_𝒪(ℂ)𝕂.

It is a finite dimensional vector space over 𝕂, and for generic z, y it has a basis given by weight functions ω^+_I, |I|=k.

 Let z_1,…,z_n,y be generic complex numbers. Then
 
    Θ̅^+(z_1,…,z_n,y)=⊕_k=0^nΘ̅^+_k(z_1,…,z_n,y).

 is a representation of the elliptic quantum group E_τ,y(𝔤𝔩_2) with the L-operator
 
    L(w)(v⊗ u)=ψ(λ-yh^(2))^(1)ℓ(w,y,λ)(ψ(λ)^-1)^(1)(v⊗τ_-μ^*u),     v∈ V_μ.

 Here ψ is the gauge transformation of Proposition <ref>.

The homomorphisms ℓ obey the RLL-type relations (<ref>) with R-matrix R^+ which, according to Proposition <ref>, is obtained from R by the gauge transformation ψ. It is easy to check that

    ℓ̂(w,y,λ)= ψ(λ-yh^(2))^(1)ℓ(w,y,λ)(ψ(λ)^-1)^(1)

obeys the same relations but with R^+ replaced by R. It follows that the corresponding difference operators define a representation of the elliptic dynamical quantum group.


 It follows from the previous section that this representation is isomorphic to the tensor product V(z_σ(1))⊗⋯⊗ V(z_σ(n)) for any permutation σ∈ S_n. However this identification with a tensor product of evaluation vector representations depends on a choice of ordering of the z_1,…,z_n, while Θ̅^+(z_1,…,z_n,y,λ) depends as a representation only on the set {z_1,…, z_n}.




 §.§ Pairing

We define a pairing of Θ^-_k with Θ^+_k, taken essentially from <cit.>. Note that the product of a function in Θ^-_k(z,y,λ) and a function in Θ^+_k(z,y,λ) divided by the products of Jacobi theta functions in part (2) of Definitions <ref> and <ref>, is a function which is doubly periodic in each variable t_i with poles at t_i=z_a and at t_i=z_a-y, a=1,…,n. It can thus be viewed as a meromorphic function on the Cartesian power E^k of the elliptic curve E=ℂ/(ℤ+τℤ).
 Let z_1…,z_n,y∈ E such that z_a≠ z_b+jy for all 1≤ a,b≤ n, 1≤ j≤ n-1, and let γ∈ H_1(E∖{z_1,…,z_n}) be the sum of small circles around z_a, a=1,…,n, oriented in counterclockwise direction. Let D⊂ E^k be the effective divisor D=∪_a=1^n∪_i=1^k({t∈ E^k t_i=z_a}∪{t∈ E^k t_i=z_a-y}). The symmetric group S_k acts by permutations on the sections of the sheaf 𝒪(D) of functions on E^k with divisor of poles bounded by D. Let ⟨ ⟩Γ(E^k,𝒪(D))^S_k→ℂ be the linear form
 
    f→⟨ f⟩= θ(y)^k/(2π i)^kk!∫_γ^k f(t_1,…,t_k)∏_1≤ i≠ j≤ kθ(t_i-t_j)/θ(t_i-t_j+y) dt_1⋯ dt_k.

For k=0 we define ⟨ ⟩ℂ→ℂ to be the identity map.

 Let f∈Γ(E^k,𝒪(D))^S_k. Then
 
    ⟨ f⟩=θ(y)^k∑_1≤ i_1<⋯<i_k≤ nres_t_1=z_i_1⋯res_t_k=z_i_k(f(t_1,…,t_k)∏_i≠ jθ(t_i-t_j)/θ(t_i-t_j+y)).


 By the residue theorem, ⟨ f⟩ is a sum of iterated residues at t_i=z_a(i) labeled by maps a [k]→[n]. Since θ(t_i-t_j) vanishes for t_i=t_j, only injective maps a contribute non-trivially. Moreover, since the integrand is symmetric under permutations of the variables t_i, maps a differing by a permutation of {1,…,k} give the same contribution. Thus we can restrict the sum to strictly increasing maps a and cancel the factorial k! appearing in the definition.

 Denote Q=∏_i=1^k∏_a=1^nθ(t_i-z_a)θ(t_i-z_a+y) and let
 
    ⟨ ,  ⟩ Θ^-_k(z,y,λ)⊗Θ^+_k(z,y,λ)→ℂ

 be the bilinear pairing ⟨ f,g⟩= ⟨ fg/Q⟩, defined for generic z∈ℂ^n, y∈ℂ. Note that fg/Q is an elliptic function of t_i for all i.

Here is the explicit formula for the pairing:

    ⟨ f,g⟩=θ(y)^k/(2π i)^kk!∫_γ^kf(t_1,…,t_k)g(t_1,…,t_k)/∏_i,aθ(t_i-z_a)θ(t_i-z_a+y)∏_i≠ jθ(t_i-t_j)/θ(t_i-t_j+y) dt_1⋯ dt_k.


 Let n=1. Then ⟨ω^-_0,ω^+_0⟩=1,
 
    ⟨ω^-_1,ω^+_1⟩=θ(λ-y) θ(λ),

 and ⟨ω^-_k,ω^+_k⟩=0 for k>1.

 The first claim holds by definition. We have
 
    ⟨ω^-_1,ω^+_1⟩ =θ(y) res_t=zθ(λ-t+z-y) θ(λ+t-z)/θ(t-z)θ(t-z+y) dt =θ(λ-y) θ(λ).

 For k≥ 2, the residue at t_1=z is regular at t_i=z for i≥2 and thus the iterated residue vanishes.

  =0pt
 
  (i) The pairing restricts to a non-degenerate pairing Θ̅^-_k(z,y,λ) ⊗Θ̅^+_k(z,y,λ) →ℂ for generic z_1,…,z_n,y,λ.
 
  (ii) In the notation of Proposition <ref>, suppose f_i∈Θ̅^-_k'_i(z',y,λ+y(n”-2k_i”)), g_i∈Θ̅^+_k”_i(z”,y,λ), i=1,2 and k_1'+k_2'=k_1”+k_2”. Then
 
    ⟨ f_1*f_2,g_1*g_2⟩= ⟨ f_1,g_1⟩⟨ f_2,g_2⟩,    if k'_1=k”_1 and k'_2=k”_2, 
    
     0,   otherwise.

 

 It is sufficient to prove (ii), since with Lemma <ref> it implies that, with a proper normalization, weight functions form dual bases with respect to the pairing.

We use Lemma <ref> to compute ⟨ f_1*f_2,g_1*g_2⟩. Let us focus on the summand in Lemma <ref> labeled by i_1<⋯<i_n and suppose i_s≤ n'<i_s+1. Due to the factor θ(t_l-z_a) in φ^-, see Proposition <ref>, the only terms in the sum over shuffles having nonzero first s residues res_t_1=z_i_1, …, res_t_s=z_i_s are those for which t_1,…, t_s are arguments of f_1. In particular the summand vanishes unless s≤ k_1'. Similarly the factors θ(t_j-z_b) in φ^+ restrict the sum over shuffles to those terms for which t_s+1,…,t_k are arguments of g_2, so that the summand vanishes unless s≥ k-k_2”=k_2'. It follows that if k_1'<k_2' then ⟨ f_1*f_2,g_1*g_2⟩ vanishes and that if k_1'=k_1”, the pairing can be computed explicitly as sum over i_1<⋯<i_s≤ n'<i_s+1<⋯<i_k, with s=k_1', of terms involving f_1g_1(z_i_1,…,z_i_k_1')f_2g_2(z_i_k_1'+1,…,z_i_k). The coefficients combine to give ⟨ f_1,g_1⟩⟨ f_2,g_2⟩.

There remains to prove that the pairing vanishes also if k_1'>k_2'. Here is where the vanishing condition comes in. We first consider the case where k_1'-k_2'=1 and then reduce the general case to this case.

As above the presence of the vanishing factors in φ^± imply that the non vanishing residues in Lemma <ref> are those labeled by i_1<⋯<i_k such that at least k_1” indices are ≥ n' and the corresponding variables t_i are arguments of g_1 and at least k_2' indices are ≤ n' and the corresponding variables are arguments of f_2. If k_1'-k_2'=1 there is one variable left and we can write the pairing as a sum of one-dimensional integrals over this variable:
 
    I_A,B=∫_γf_1(z_A,t)g_1(z_B)f_2(z_A)g_2(t,z_B)/h(z_1,…,z_n,y,t) dt.

Here z_A=z_a_1,…,z_a_k_2' with a_i≤ n' and z_B=z_b_1,…,z_b_k_1” with b_i>n'. The point is that in h(z,t) several factor cancel and one obtains

    h(z,y,t)=C(z,y)∏_c∈A∪ Bθ(t-z_c) ∏_c∈ A∪ Bθ(t-z_c+y),

for some t-independent function C(z,y). Because of the vanishing condition, the integrand in (<ref>) is actually regular at t=z_c-y and the only poles are at t=z_c, c∈A∪ B. By the residue theorem I_A,B=0.

Finally, let us reduce the general case to the case where k_1'-k_2'=1. We use induction on n. By Lemma <ref> the pairing vanishes unless k=1,0 so there is nothing to prove in this case. Assume that the claim is proved for n-1. By Proposition <ref>, we can write g_1=h_1*m_1 and g_2=h_2*m_2 with m_i∈Θ̅^-_r_i(z_n,y,λ). By Lemma <ref> we can assume that r_i∈{0,1}. By the associativity of the shuffle product we can use the result for k_1'-k_2'=1 to obtain that the pairing vanishes unless r_1=r_2 and

    ⟨ f_1*g_1,f_2*g_2⟩=⟨ f_1*h_1,f_2*h_2⟩⟨ m_1,m_2⟩.

By the induction hypothesis, this vanishes unless k_1'=k_2'.


We obtain orthogonality relations for weight functions. To formulate them we introduce some notation. For I⊂ [n] and 1≤ j≤ n we set

    n(j,I)=|{l∈[n] | l∈ I, l>j}|,
    
     w(j,I)=n(j,I)-n(j,I̅).

Thus -w(j,I) the sum of the weights of the tensor factors to the right of the j-th factor in v_I.
 [cf. <cit.>]
 
    ⟨ω^-_I,ω^+_J⟩ = δ_I,Jψ_I(y,λ),

 where ψ_I(y,λ)= ∏_j∈ Iθ(λ-(w(j,I)+1)y)θ(λ-w(j,I)y).
 

 

 §.§ Normalized weight functions

 By construction the weight functions ω^±_I are entire functions of all variables and obey the vanishing conditions
 
    ω^±_I(z_a,z_a-y,t_3,…,t_k;z,y,λ)=0,      a=1,…,n.

 This motivates the following definition.
 
 The normalized weight functions w_I^± are the functions
 
    w_I^-(t;z,y,λ) =ω^-_I(t;z,y,λ)/∏_1≤ j≠ l≤ kθ(t_j-t_l+y),
    
     w_I^+(t;z,y,λ) = ω^+_I(t;z,y,λ)/ψ_I(y,λ) ∏_1≤ j≠ l≤ kθ(t_j-t_l+y).

 
  The factor 1/ψ_I, defined in Corollary <ref>, simplifies the orthogonality relations and the action of the permutations of the z_i at the cost of introducing poles at λ+yℤ modulo
 ℤ+τℤ.
 
 Let I={i_1,…,i_k}⊂[n] and f(t_1,…,t_k) a symmetric function of k variables. We write f(z_I) for f(z_i_1,…,z_i_k).
  For each I,J⊂[n] such that |I|=|J|, the weight functions w_I^-(z_J;z,y,λ) and ψ_I(y,λ) w_I^+(z_J;z,y,λ) are entire functions of z, y, λ.
 
  The vanishing condition implies that ω_I^±(z_a,z_b,t_3,…) is divisible by θ(z_b-z_a+y) so that the quotient by θ(t_2-t_1+y) is regular at z_b=z_a-y after substitution t_1=z_a, t_2=z_b. Since ω_I^± is a symmetric function, the same holds for any other pair t_j, t_l.
 
 The orthogonality relations become:
 [cf. <cit.>] Let I,J⊂[n], |I|=|J|=k. The normalized weight functions obey the orthogonality relations
 
    ∑_Kw^-_I(z_K,z,y,λ)w^+_J(z_K,z,y,λ)/∏_a∈ K∏_b∈K̅θ(z_a-z_b)θ(z_a-z_b+y) =δ_I,J.

 The summation is over subsets K⊂[n] of cardinality |K|=k.

 This is a rewriting of Corollary <ref> by using Lemma <ref>.


We will also use the orthogonality relations in the following equivalent form.
  Let I,K⊂ [n], |I|=|K|=k. We have

    ∑_J w^-_J(z_I,z,y,λ) w_J^+(z_K,z,y,λ)=
    ∏_a∈ I, b∈I̅θ(z_a-z_b)θ(z_a-z_b+y),   I=K,
    
    0,   otherwise.


Let

    x_IK =w^-_I(z_K,z,y,λ)/∏_a∈ K,   b∈K̅θ(z_a-z_b),     
    y_KJ =w^+_J(z_K,z,y,λ)/∏_a∈ K,   b∈K̅θ(z_a-z_b+y).

Proposition <ref> claims that the matrix (x_IK)_I,K is the left inverse of the matrix (y_KJ)_K,J. This implies, however, that the matrix (x_IK)_I,K is also a right inverse of (y_KJ)_K,J, which is equivalent to the statement of the corollary.
 

Weight functions have a triangularity property. Introduce a partial ordering on the subsets of [n] of fixed cardinality k: if I={i_1<…<i_k} and J={j_1<…<j_k}, then I≤ J if and only if i_1≤ j_1, …, i_k≤ j_k.
 Let ϵ[n]^2→{0,1} be such that
 
    ϵ(a,b)=1,   if a>b,
    
     0,   if a<b.

 Then
 =0pt
 
  (i) w_I^-(z_J;z,y,λ) vanishes unless J≤ I and
 
    w_I^-(z_I;z,y,λ)=∏_a∈ Iθ(λ-(w(a,I)+1)y)∏_a∈ I,  b∈I̅θ(z_a-z_b+ϵ(b,a)y).

 
  (ii) w_I^+(z_J,z,y,λ) vanishes unless I≤ J and
 
    w_I^+(z_I;z,y,λ)=∏_a∈ I, b∈I̅θ(z_a-z_b+ϵ(a,b)y)/∏_a∈ Iθ(λ-(w(a,I)+1)y).

 



 §.§ Eigenvectors of the Gelfand–Zetlin algebra

The normalized weight functions w^-_I evaluated at z_J provide the (triangular) transition matrix between the standard basis of (ℂ^2)^⊗ n and a basis of eigenvectors of the Gelfand–Zetlin
algebra. The Gelfand–Zetlin algebra is generated by the determinant Δ(w), see (<ref>), and L_22(w). The determinant acts by multiplication by

    ∏_i=1^nθ(w-z_i+y)/θ(w-z_i).

 We thus need to diagonalize L_22(w).
 Let 0≤ k≤ n, [k]={1,…,k}. Then
 
    ξ_[k]= ∏_i=1^kθ(λ+(n-k-i)y)v_[k]∈ V(z_1)⊗⋯⊗ V(z_n)

 is an eigenvector of L_22(w) with eigenvalue
 
    ∏_a=1^kθ(w-z_a)/θ(w-z_a-y).


 (See <cit.>.) Since L_21(w)v_1=0=L_12(w)v_2, the action of L_22(w) on v_1^⊗ k⊗ v_2^⊗ n-k is simply the product of the action on all factors, with the appropriate shift of λ. Since L_22(w) acts diagonally in the basis v_1, v_2 one gets the result by straightforward calculation.


For I⊂[n], |I|=k, define

    ξ_I=ξ_I(z,y,λ)= ∑_|J|=kw^-_J(z_I,z,y,λ)/∏_a∈ I,  b∈I̅θ(z_a-z_b+y) v_J.

By Lemma <ref> this definition is consistent with the one for
ξ_[k] above.
[cf. <cit.>] The vectors ξ_I, I⊂[n], |I|=k form a basis of eigenvectors of the operators of the Gelfand–Zetlin algebra on V(z_1)⊗⋯⊗ V(z_n):
 
    Δ(w)ξ_I=∏_a=1^nθ(w-z_a+y)/θ(w-z_a)ξ_I,      L_22(w)ξ_I= ∏_a∈ Iθ(w-z_a)/θ(w-z_a-y)ξ_I.



 By Corollary <ref>(i), we have that
 
    ξ_I(z,y,λ) =S_i(z,y,λ)ξ_s_i· I(s_iz,y,λ).

Thus ξ_I(z,y,λ) is related to ξ_s_i· I(s_iz,y,λ) by a morphism of representations of the elliptic dynamical quantum group. If |I|=k then there is a permutation σ such that σ· I=[k] and thus ξ_I(z,y,λ)=ρ(σ) ξ_[k](σ· z,y,λ) for some morphism ρ(σ). Since ξ_[k](z,y,λ) is an eigenvector of L_22(w) with eigenvalue μ_[k](w;z,y), see Lemma <ref>, we deduce that ξ_I(z,y,λ) is an eigenvector with eigenvalue μ_I(w;z,y)=μ_[k](w;σ· z,y).



 §.§ An explicit formula for weight functions

Let I={i_1<i_2<⋯<i_k}⊂ [n] and recall the definition (<ref>) of w(i,I) and of ψ_I in Corollary <ref>. We set

    l^+_I(r,a;t,z,y,λ) =
     θ(t_r-z_a+y),   if a<i_r, 
    θ(λ+t_r-z_a-w(i_r,I)y),    if a=i_r, 
    θ(t_r-z_a),    if a>i_r,

and

    l^-_I(r,a;t,z,y,λ) =
     θ(t_r-z_a),   if a<i_r, 
    θ(λ-t_r+z_a-(w(i_r,I)+1)y),    if a=i_r, 
    θ(t_r-z_a+y),    if a>i_r.

Then

    w^+_I(t;z,y,λ)=1/ψ_I(y,λ)Sym( ∏_r=1^k∏_a=1^nl^+_I(r,a;t,z,y,λ) /∏_1≤ i<j≤ kθ(t_i-t_j)θ(t_j-t_i+y)),

and

    w^-_I(t;z,y,λ)=Sym( ∏_r=1^k∏_a=1^nl^-_I(r,a;t,z,y,λ) /∏_1≤ i<j≤ kθ(t_j-t_i)θ(t_i-t_j+y)).



 §.§ Dual bases and resonances

Here we prove Propositions <ref> and <ref>. They are corollaries of the following more precise statement:
 Let Λ=ℤ+τℤ and fix (z,y,λ)∈ℂ^n×ℂ×ℂ. Assume that z_a-z_b-sy∉Λ, for all a≠ b, s=0,…,k, λ-sy∉Λ for all s∈ S⊂ℤ for some finite S depending on k and n.
 =0pt
 
  (i) Let ω^±_k_a=ω_k_a^±(t;z_a,λ+y∑_b=a+1^n(1-2k_b)), see Example <ref>. Then the family
 
    ω^±_k_1*…*ω^±_k_n,      k∈ℤ^k_≥0,    ∑_a=1^n k_a=k

 is a basis of Θ^±_k(z,y,λ).
 
  (ii) The subfamily indexed by (k_a)_a=1^n such that k_a∈{0,1} for all a is a basis of Θ̅^±_k(z,y,λ).
 

Part (i) and a special case of (ii) are proved in <cit.>. The proof relies on the following construction of linear forms whose evaluations on the members of the family form a non-degenerate triangular matrix. For a symmetric function f(t_1,…,t_k) and w∈ℂ, define ev_wf to be the symmetric function of k-1 variables

    ev_wf(t_1,…,t_k-1)=f(t_1,…,t_k-1,w).

It is easy to check that ev_w maps Θ^±_k(z,y,λ) to Θ^±_k-1(z,y,λ± y). For ℓ∈ℤ_≥0 and f a symmetric function of k variables set

    ev_w,ℓ(f)=
     f   if ℓ=0,
    ev_w-ℓ y∘⋯∘ev_w-y∘ev_w(f)    if 1≤ℓ≤ k,
    
     0,   otherwise.

Finally, for ℓ∈ℤ_≥0^n, ∑_a=1^nℓ_a=k we introduce linear forms ϵ_ℓ_1,…,ℓ_n∈Θ^±(z,y,λ)^*:

    ϵ_ℓ_1,…,ℓ_n= ev_z_n,ℓ_n∘⋯∘ev_z_1,ℓ_1.

The following result is a special case of Proposition 30 from <cit.> (adapted to the conventions of this paper). It can be checked by induction using the fact that the evaluation points are such that at most one shuffle in the definition of the shuffle products contributes nontrivially.
 Let (z,y,λ)∈ℂ^n×ℂ×ℂ, k,ℓ∈ℤ_≥0^n with ∑_a=1^nℓ_a= ∑_a=1^n k_a. Then
 =0pt
 
  (i) Let f_a∈Θ^-_k_a(z_a,y,λ-y∑_b=a+1^n(2k_b-1)), a=1,…,n. Then
 
    ϵ_ℓ_1,…,ℓ_n(f_1*⋯*f_n)=0,

 unless ℓ_1+⋯+ℓ_p≤ k_1+⋯+k_p for all p=1,…,n, and
 
    ϵ_k_1,…,k_n (f_1*⋯*f_n) 
         =∏_a=1^nev_z_a,k_a(f_a) ∏_a<b(∏_s=0^k_b-1θ(z_b-z_a-ys) ∏_s=0^k_a-1θ(z_a-z_b+y(1-s))).

 
  (ii) Let ω^-_k be the basis of Θ^-_k(z,y,λ), z∈ℂ, defined in Example <ref>. Then
 
    ev_z,kω^-_k=∏_s=1^kθ(λ-sy).

 


Setting f_a=ω^-_k_a, a=1,…, n gives a proof of Proposition <ref> (i) in the case of Θ^-. The case of Θ^+ is reduced to this case by Proposition <ref>.

We turn to the proof of Proposition <ref>(ii). In the notation we have introduced here, Θ̅^±_k(z,y,λ) is the intersection of the kernels of ev_z_a,2 for a=1,…,n.

Let (z,y,λ)∈ℂ^n×ℂ×ℂ and 1≤ c≤ n. In the following proposition we describe the interaction of ev_z_c and ev_z_c-y with the shuffle product. By using the identifications of Remark <ref>, we view these maps as maps between the following spaces:

    ev_z_c Θ^±_k(z,y,λ)→Θ^±_k-1(z_1,…,z_c-y,…,z_n,y,λ),
    ev_z_c-y Θ^±_k(z,y,λ)→Θ^±_k-1(z_1,…,z_c+y,…,z_n,y,λ± 2y).

 In the notation of Proposition <ref>, let f∈Θ^-_k'(z',y,λ+y(n”-2k”)), g∈Θ^-_k”(z”,y,λ). We have
=0pt

  (i) ev_z_c(f*g)= ev_z_c(f)*g ∏_b=n'+1^nθ(z_c-z_b+y), 1≤ c≤ n',

  (ii) ev_z_c-y(f*g)= f*ev_z_c-y(g) ∏_a=1^n'θ(z_c-z_a-y), n'<c≤ n.

Similarly, let f∈Θ^+_k'(z',y,λ+y(n”-2k”)), g∈Θ^+_k”(z”,y,λ). We have
=0pt

  (iii) ev_z_c-y(f*g)= ev_z_c-y(f)*g ∏_b=n'+1^nθ(z_c-z_b+y), 1≤ c≤ n',

  (iv) ev_z_c(f*g)= f*ev_z_c(g) ∏_a=1^n'θ(z_c-z_a-y), n'<c≤ n.


 (i) Due to the factor ∏_l=k'+1^kθ(t_l-z_c) in the definition of φ^-, see Proposition <ref>, the only terms in the sum over permutations contributing nontrivially to ev_z_c(f*g) are such that t_k is an argument of f. Thus ev_z_c(f*g)=ev_z_c(f)*g times a factor that is computed explicitly. The proof of (ii)–(iv) is similar.


By iterating, we obtain:
 Let f∈Θ^-_k'(z',y,λ+y(n”-2k”)), g∈Θ^-_k”(z”,y,λ). We have
 
    ev_z_c,2(f*g)= ev_z_c,2(f)*g ∏_b=n'+1^nθ(z_c-z_b+y)θ(z_c-z_b),      1≤ c≤ n', 
    ev_z_c,2(f*g) = f*ev_z_c,2(g) ∏_a=1^n'θ(z_c-z_a-y)θ(z_c-z_a),      n'<c≤ n,

 and similarly for Θ^+.
 In particular, if f and g satisfy the vanishing condition then
 also f*g does.


 We give the proof for Θ̅^-_k(z,y,λ). The proof for Θ̅^+_k(z,y,λ) is similar or can be deduced using the duality map of Proposition <ref>. It follows from Corollary <ref> that the indicated subfamily does belong to Θ̅^-_k(z,y,λ). It remains to show that it is a spanning set. By Proposition <ref>, (i), we know that any element of Θ̅^-_k(z,y,λ) can be written as linear combinations

    ∑_k_1+⋯+k_n=kλ_k_1,…,k_nω_k_1*⋯*ω_k_n.

On the other hand, the linear form ϵ_k_1,…,k_n vanishes on Θ̅^-_k(z,y,λ) if k_a≥2 for some a, since it involves the evaluation at z_a,z_a-y. By Lemma <ref>, the coefficients λ_k_1,…,k_n must thus vanish if at least one k_a is ≥2 which is what we had to prove.




§ EQUIVARIANT ELLIPTIC COHOMOLOGY OF GRASSMANNIANS


Let E be an elliptic curve and G a compact group. Equivariant elliptic cohomology was postulated by Ginzburg, Kapranov and Vasserot in <cit.> as a functor E_G from pairs of finite G-CW complexes to superschemes satisfying a set of axioms, generalizing those satisfied by equivariant cohomology and equivariant K-theory. One of them being that for a point pt, E_G(pt) is a suitable moduli scheme of G-bundles on the dual elliptic curve. For example E_U(n)(pt)=E^(n)=E^n/S_n and for an abelian group A with group of characters X(A)=Hom(A,U(1)), E_A(pt)=Hom(X(A),E). By functoriality, the scheme E_G(M) comes with a structure map

    p_G E_G(M)→ E_G(pt).

For a complex elliptic curve, the case we consider here, a construction of equivariant elliptic cohomology was given by Grojnowski <cit.>. It has the property that for a connected Lie group G with maximal torus A and Weyl group W then W acts on E_A(M) and E_G(M)=E_A(M)/W.



 §.§ Tautological bundles and Chern classes

Let E be a complex elliptic curve. The unitary group U(n) and its maximal torus A≅ U(1)^n act on the Grassmannian Gr(k,n) of k-dimensional subspaces of ℂ^n. The A-equivariant cohomo­logy of Gr(k,n) was described in <cit.>, and is analogous to the classical description of ordinary equivariant cohomology in terms of Chern classes of tautological bundles. The Grassmannian has two tautological equivariant vector bundles of rank k and n-k, respectively. They give rise to a characteristic map <cit.>

    χ E_A(Gr(k,n))→ E_U(k)(pt)× E_U(n-k)(pt)=E^(k)× E^(n-k).
 Here E^(k)=E^k/S_k denotes the symmetric power of the elliptic curve, which is the U(k)-equivariant cohomology of a point. Together with the structure map to E_A(pt) we have a description of
the equivariant elliptic cohomology as the fiber product of E^(k)× E^(n-k) and E^n over E^(n), namely, we have the Cartesian square:

    [   E_A(Gr(k,n))              ⟶ E^(k)× E^(n-k);              ↓                             ↓;            E^n              ⟶         E^(n). ]

The left vertical arrow is the structure map to E_A(pt); the maps E^n→ E^(n) and E^(k)× E^(n-k)→ E^(n) are the canonical projections. Thus E_A(Gr(k,n))=(E^(k)×
E^(n-k))×_E^(n)E^n.

The symmetric group S_n (the Weyl group of U(n)) acts on the diagram above (with trivial action on the right column) and the U(n)-equivariant cohomology is the quotient by this action:

    E_U(n)(Gr(k,n))=E_A(Gr(k,n))/S_n=E^(k)× E^(n-k).




 §.§ Moment graph description
 An alternative useful description of the equivariant elliptic cohomology is via the localization theorem, proved by Goresky, Kottwitz and MacPherson <cit.> for equivariant cohomology and generalized to elliptic cohomology by Knutson and Roşu <cit.>. For partial flag varieties such as Grassmannians it is described explicitly in <cit.>. The action of A on the Grassmannian
Gr(k,n) has isolated fixed points labeled by subsets of [n]={1,…,n} with k elements. The fixed point x_I labeled by I⊂[n] is the k-plane spanned by the coordinate axes indexed by I. The inclusion of the fixed points x_I induces a map ι_I E_A(pt)→ E_A(Gr(k,n)) and it turns out that E_A(Gr(k,n)) is the union of the ι_IE_A(pt)≃ E^n where I runs over the subsets of [n] with k elements.

Let Γ be the graph with vertex set Γ_0 the set of subsets I⊂ [n] with |I|=k elements and an edge connecting I with I' for each pair of vertices such that |I∩ I'|=k-1. In this case I=J∪{a} and I'=J∪{b} with |J|=k-1 and we set Δ_I,I'={z∈ E^n, z_a=z_b}. We then have inclusion maps Δ_I,I'→ι_IE_A(pt)=E^n, Δ_I,I'→ι_I'E_A(pt)=E^n.
 We have the coequalizer diagram
 
    ⊔_|I∩ I'|=k-1Δ_I,I'⇉⊔_I⊂ [n],|I|=kE^n→ E_A(Gr(k,n))).


In other words, E_A(Gr(k,n)) is the union of copies of E^n labeled by subsets I⊂ [n] of size k, glued along the diagonals Δ_I,I'. The structure map E_A(Gr(k,n))→ E^n is the identity on each copy. The isomorphism between the two descriptions of E_A(Gr(k,n)) is induced by the map

    ⊔_I⊂ [n],|I|=kE^n→(E^(k)× E^(n-k))×_E^(n)E^n,

whose restriction to the copy E^n labeled by I is

    z↦ (z,z_I,z_I̅),     z_I=(z_i)_i∈ I,      z_I̅=(z_j)_j∈I̅.

It is easy to check directly Proposition <ref> using the fiber product as a definition of the equivariant elliptic cohomology.


 §.§ Cotangent bundles and dynamical parameter

The action of U(n) on the Grassmannian induces an action on its cotangent bundle X_k,n=T^*Gr(k,n). Additionally we have an action of U(1) on the cotangent bundle by scalar multiplication on the fibers, so we get an action of

    G=U(n)× U(1)

and its Cartan torus

    T=A× U(1)≅ U(1)^n+1.

Since the cotangent bundle is equivariantly homotopy equivalent to its zero section, the equivariant elliptic cohomology is simply

    E_T(X_k,n)=E_A(Gr(k,n))× E,

a scheme over E_T(pt)=E^n× E, and

    E_G(X_k,n)=E_U(n)(Gr(k,n))× E,

a scheme over E_G(pt)=E^(n)× E.

We will consider, as in <cit.>, an extended version of elliptic cohomology to accommodate for dynamical variables in quantum group theory, namely

    Ê_T(X_k,n):=E_T(X_k,n)× (Pic(X_k,n)⊗_ℤE)≅ E_T(X_k,n)× E,

a scheme over Ê_T(pt)=E^n× E× E (the Picard group of the Grassmannian is infinite cyclic generated by the top exterior power of the tautological bundle). Similarly, we set

    Ê_G(X_k,n)=E_G(X_k,n)× E,

which is a scheme over Ê_G(pt)=E^(n)× E× E.

The fixed points x_K for the A-action on the Grassmannian are also isolated fixed points in the cotangent bundle of the Grassmannian for the T-action and we have maps ι_K=Ê_T(i_K)Ê_T(pt) →Ê_T(X_k,n) induced by the inclusion i_Kpt↦ x_K. Then Ê_T(X_k,n) consists of the components ι_KÊ_T(pt), where K runs over the subsets of [n] with k elements. By Section <ref>, we have a description of Ê_T(X_k,n) as a fiber product:

    Ê_T(X_k,n)≅(E^(k)× E^(n-k))×_E^(n)E^n× E× E.

In particular we have the characteristic embedding

    c Ê_T(X_k,n)→ E^(k)× E^(n-k)× E^n× E× E

of the extended T-equivariant elliptic cohomology scheme into a non-singular projective variety.



§ ADMISSIBLE LINE BUNDLES ON Ê_T(X_K,N)



 §.§ Line bundles on E^p

Line bundles on complex tori are classified by the Appel–Humbert theorem, see <cit.>. In the special case of powers of generic elliptic curves this reduces to the following explicit description: let E=ℂ/Λ with Λ=ℤ+τℤ so that E^p=ℂ^p/Λ^p. For each pair (N,v) consisting of a symmetric integral p× p matrix N and v∈(ℂ/ℤ)^p let ℒ(N,v) be the line bundle  with action

    λ·(x,u)=(x+λ,e_λ(x)u),    λ∈Λ^p,      x∈ℂ^p,     u∈ℂ,

and cocycle

    e_n+mτ(x)=(-1)^n^tNn(- e^ iπτ)^m^tNm e^2π i m^t(Nx+v),      n,m∈ℤ^p.

    =0pt
 
  (i) ℒ(N,v) is isomorphic to ℒ(N',v') if and only if N=N' and v≡ v'Λ^p.
 
  (ii) For generic E, every holomorphic line bundle on E^p is isomorphic to ℒ(N,v) for some (N,v).
 
  (iii) ℒ(N_1,v_1)⊗ℒ(N_2,v_2)≅ℒ(N_1+N_2,v_1+v_2).
 
  (iv) Let σ∈ S_p act by permutations on E^p and ℂ^p. Denote also by σ the corresponding p× p permutation matrix. Then
 
    σ^*ℒ(N,v)=ℒ(σ^t Nσ,σ^t v).

 


To an integral symmetric p× p matrix N and a vector v∈ℂ^p we associate the integral quadratic form N(x)=x^tNx and the linear form v(x)=x^tv on the universal covering ℂ^p of E^p and we call them the quadratic form and the linear form of the line bundle ℒ(N,v). The linear form is defined up to addition of an integral linear form.

 Exceptions to (ii) are elliptic curves with complex multiplication, in which case there are additional line bundles that are not isomorphic to those of the form ℒ(N,v).

 The map E^p→Pic(E^p) sending v to ℒ(0,v) is an isomorphism onto the subgroup Pic^0(E^p) of classes of line bundles of degree 0. If E is a generic elliptic curve, the Néron–Severi group NS(E^p)=Pic(E^p)/Pic^0(E^p) is a free abelian group of rank n(n+1)/2 identified with the group of integral symmetric matrices via N↦ℒ(N,0).

  Sections of ℒ(N,v) are the same as functions f on ℂ^p such that f(x+λ)=e_λ(x)^-1f(x) for all λ∈Λ^p, x∈ℂ^p. Explicitly, a function on ℂ^p defines a section of ℒ(N,v) if and only if
 
    f(x_1,…,x_j+1,…,x_p) = (-1)^N_jjf(x), 
    
     f(x_1,…,x_j+τ,…,x_p) = (-1)^N_jj e^-2π i(∑_k N_jkx_k+v_j)-π iτ N_jj f(x),
 for all x∈ℂ^p, j=1,…,p.

 The factors of -1 in the cocycle can be removed by going to an equivalent cocycle. With the present convention the line bundles ℒ(N,0) correspond to divisors whose irreducible components are subgroups. Let θ(z) be the odd Jacobi theta function in one variable, see (<ref>). Then, for any r∈ℤ^p and z∈ℂ,
 
    θ(r^tx+z)=θ(r_1x_1+⋯+r_px_p+z)

 is a holomorphic section of ℒ(N,v) with quadratic form
 
    N(x)=(∑_i=1^pr_ix_i)^2,

 and linear form
 
    v(x)=z∑_i=1^p r_ix_i.

If z=0 this section vanishes precisely on the subtorus Ker(ϕ_r), the kernel of the group homomorphism ϕ_r E^p→ E, x↦∑_ir_ix_i. Since an integral quadratic form is an integral linear combination of squares of integral linear forms, ℒ(N,0) has a meromorphic section which is a ratio of products of theta functions θ(r^tx) with r∈ℤ^p.



 §.§ Admissible line bundles

The elliptic dynamical quantum group acts on sections of admissible line bundles, which are, up to a twist by a fixed line bundle, those coming from the base scheme Ê_T(pt). Let p_T be the structure map

    p_T Ê_T(X_k,n)→Ê_T(pt),

and χ̂=χ×id×id the characteristic map

    χ̂ Ê_T(X_k,n)=E_A(Gr(k,n))× E× E→ E^(k)× E^(n-k)× E× E.

Let t_1,…,t_k,s_1,…,s_n-k,y,λ be coordinates on the universal covering of E^k× E^n-k× E× E and N the quadratic form

    N_k,n(t,s,y,λ)=2∑_i=1^k t_i(λ+(n-k)y)+∑_i=1^k∑_j=1^n-k(t_i-s_j)^2.

Clearly N_k,n is symmetric under permutation of the coordinates t_i and of the coordinates s_j and thus ℒ(N_k,n,0) can be considered as a bundle on E^(k)× E^(n-k)× E× E.

 The twisting line bundle on X_k,n is 𝒯_k,n=χ̂^*ℒ(N_k,n,0)

 An admissible line bundle on Ê_T(X_k,n) is a line bundle of the form
 
    p_T^*ℒ⊗𝒯_k,n,

 for some line bundle ℒ on Ê_T(pt).



 §.§ Holomorphic and meromorphic sections

We will consider meromorphic sections of line bundles on elliptic cohomology schemes. Since these schemes are singular, we need to be careful about the definition. Recall that Ê_T(X_k,n) has components Y_K=ι_KÊ_T(pt), corresponding to the inclusion of the fixed points x_K, labeled by subsets K⊂[n] of cardinality k. We say that a meromorphic section on a complex manifold restricts to a meromorphic section on a submanifold if it is defined at its generic point, i.e., if the divisor of poles does not contain a component of the submanifold.
 Let ℒ be a line bundle on Ê_T(X_k,n). A meromorphic section of ℒ is a collection of meromorphic sections s_I of ℒ|_Y_I, labeled by I⊂[n] with |I|=k and restricting to meromorphic sections on all intersections Y_I_1∩⋯∩ Y_I_s and such that
 
    s_I|_Y_I∩ Y_J=s_J|_Y_I∩ Y_J,

for all I, J. A holomorphic section is a meromorphic section whose restriction to each Y_I is holomorphic. We denote by Γ(Ê_T(X_k,n),ℒ) the space of holomorphic sections of ℒ and by Γ_mer(Ê_T(X_k,n),ℒ) the space of meromorphic sections of ℒ.




 §.§ Weight functions and admissible line bundles

With the description of line bundles of Section <ref>, the weight functions w^+_I(t_1,…,t_k,z_1,…,z_n, y,λ) can be viewed as sections of certain line bundles on E^(k)× E^(n-k)× E^n× E× E, namely as (s-independent) functions of the coordinates (t_1,…,t_k,s_1,…,s_n-k,z,y,λ) on the universal covering space, with proper multipliers under lattice translations. Their pull-back by the characteristic embedding

    c  Ê_T(X_k,n)↪ E^(k)× E^(n-k)× E^n× E× E,

see (<ref>), is a section of the pull-back bundle and its restriction to ι_JÊ_T(pt) is the evaluation of w^+_I at t=z_I.

 Let I⊂[n], |I|=k. Then the restriction c ^*w^+_I of w^+_I to Ê_T(X_k,n) is a meromorphic section of the admissible bundle p_T^*ℒ_I⊗𝒯_k,n for some line bundle ℒ_I on Ê_T(pt). Moreover ψ_Ic ^*w^+_I is holomorphic.

 We need first to check that all terms in the sum over S_k defining the symmetrization map Sym in Section <ref> have the same transformation properties under shifts of the variables by the lattice so that they define sections of the same line bundle on E^k× E^n× E^2. The symmetrization map then produces a section symmetric under permutations of t_i, which is the same as a section of a line bundle on
 E^(k)× E^n× E^2. The transformation properties are encoded in the quadratic form: the argument of Sym is a section of the line bundle ℒ(M_I,0) with
 
    M_I(t,z,y,λ) =2∑_r=1^k t_r(λ+(n-k)y)+∑_r=1^k∑_a=1^n (t_r-z_a)^2-2∑_1≤ r<s≤ k(t_r-t_s)^2 + M̃_I(z,y,λ),

where M̃_I is independent of t_1,…,t_k. Since M_I is symmetric under permutations of the variables t_i it defines an S_n-equivariant line bundle. All terms in the sum over permutations are sections of this line bundle and their sum is a symmetric section, i.e., the pull-back of a section on the quotient E^(k)× E^n× E^2, which we understand as a section on E^(k)× E^(n-k)× E^n× E^2, constant along E^(n-k). The restriction to the component ι_KÊ_T(pt) of Ê_T(X_k,n) is w^+_I(z_K,z,y,λ), the result of substituting the variables t_i by z_K=(z_i)_i∈ K. It is a section of the line bundle with quadratic form
 
    M_I(z_K,z,y,λ) = 2∑_i∈ Kz_i(λ+(n-k)y) +∑_i∈ K∑_a=1^n(z_i-z_a)^2 -∑_i,j∈ K(z_i-z_j)^2+M̃_I(z,y,λ)
     =2∑_i∈ Kz_i(λ+(n-k)y) +∑_i∈ K,  j∈K̅(z_i-z_j)^2 +M̃_I(z,y,λ)
    = N_k,n(z_K,z_K̅,y,λ)+M̃_I(z,y,λ),

 cf. (<ref>). Thus the symmetrization is a section of the tensor product of the twisting bundle and the bundle with quadratic form M̃_I which is independent of K and thus comes from Ê_T(pt)=E^n× E^2. The section ψ_Ic ^*w^+_I is holomorphic because of Lemma <ref>.


Thus c ^*w_I^+ is a meromorphic section of an admissible line bundle p_T^*ℒ_I⊗𝒯_k,n with poles on a finite set of hypertori with equation λ-jy=0, j∈ℤ, the divisors of zeros of ψ_I. The bundle ℒ_I can be calculated: ℒ_I=ℒ(N_I,0) with

    N_I=-2∑_a∈I̅n(a,I)z_ay-2∑_a∈ Iz_a(λ+n(a,I̅)y)
     +(k(n-k)-∑_a∈ In(a,I̅))y^2-∑_a∈ I(λ- (n(a,I)+1)y+n(a,I̅)y)^2,

see (<ref>) for the definition of n(a,I). Let D_I be the divisor of zeros of the section ψ_I(y,λ) on Ê_T(pt), I⊂ [n], see Corollary <ref>. Then the normalized weight function can be understood as a holomorphic section of an admissible bundle:

    c ^*w^+_I∈Γ(X_k,n, p_T^*ℒ_I(D_I)⊗𝒯_k,n).

Here the notation ℒ(D) means as usual the invertible sheaf of meromorphic sections of a sheaf L whose poles are bounded by the divisor D.
 Let n=1. Then w^+_∅(t,z,y,λ)=1 and c ^*w_∅=1 is a section of the trivial bundle (ℒ_∅ and 𝒯_0,1 are both trivial). For k=1,

    w^+_{1}(t,z,y,λ)=θ(λ+t-z)/θ(λ)θ(λ-y),

and c ^*w^+_{1} is obtained by substituting t=z:

    c ^*w^+_{1}(z,y,λ)=1/θ(λ-y).

This is a meromorphic section of the line bundle ℒ(-(λ-y)^2,0) with a simple pole at λ=y on Ê_T(X_1,1)=Ê_T(pt)≅ E^3. The quadratic form is composed from the quadratic forms 2zλ of 𝒯_1,1 and -2zλ-(λ-y)^2 of ℒ_{1}.



 §.§ Elliptic cohomology classes and stable envelope

Here we introduce an elliptic version of the Maulik–Okounkov stable envelope. It is constructed in terms of weight functions. In Appendix <ref> we give an axiomatic definition in the spirit of <cit.>. It would be interesting to understand the relation of our definition with the one sketched in <cit.>.

 Let ℒ∈Pic(Ê_T(pt)). A T-equivariant elliptic cohomology class on X_k,n of degree ℒ is a holomorphic section of the admissible bundle p_T^*ℒ⊗𝒯_k,n on Ê_T(X_k,n). We denote by H_T^ell(X_k,n)_ℒ the vector space of T-equivariant elliptic cohomology classes of degree ℒ on X_k,n. We denote by H_T^ell(X_n)_ℒ the 𝔥-module ⊕_k=0^n H_T^ell(X_k,n)_ℒ, with k-th direct summand of 𝔥-weight -n+2k.

 The stable envelope is the map

    Stab (ℂ^2)^⊗ n→⊕_k=0^n⊕_I⊂ [n], |I|=k H_T^ell(X_k,n)_ℒ_I(D_I),

sending v_I to the cohomology class c ^*w_I^+.

 The basis vector v_I should be viewed as the generator of the space of elliptic cohomology classes of the fixed point x_I, see Section <ref> below.

 The class c ^*w_I^+ has analogs in equivariant cohomology and equivariant K-theory of X_k,n, see <cit.>. The analog of c ^*w_I^+ in equivariant cohomology is the equivariant Chern–Schwartz–MacPherson class of the open Schubert variety Ω_I, see <cit.>. Hence c ^*w_I^+ may be considered as an elliptic equivariant version of the Chern–Schwartz–MacPherson class.




 §.§ Sheaf of elliptic cohomology classes and theta functions

Here we realize elliptic cohomology classes as sections of coherent sheaves on Ê_T(pt) and relate their sections to the theta functions with vanishing condition of Section <ref>.
 Let k=0,…,n, μ=-n+2k and 𝒯_k,n be the twisting line bundle of Definition <ref>. The sheaf of elliptic cohomology classes of weight μ is the sheaf
 
    ℋ_T^ell(X_k,n)=p_T*𝒯_k,n

 on Ê_T(pt). Here p_T*=(p_T)_* denotes the direct image by the structure map p_TÊ_T(X_k,n)→Ê_T(pt).

By the projection formula, ℒ⊗ p_T*𝒯_k,n≅ p_T*(p_T^*ℒ⊗𝒯_k,n) for any line bundle ℒ∈Pic(Ê_T(pt)). Thus a section of ℋ_T^ell(X_k,n)⊗ℒ on an open set U is a section of the admissible line bundle p_T^*ℒ⊗𝒯_k,n on p_T^-1(U). In particular,

    H^ell_T(X_k,n)_ℒ=Γ(Ê_T(pt),ℋ^ell_T(X_k,n)⊗ℒ).


The space Θ^+_k(z,y,λ) of theta functions introduced in Section <ref> is the fiber of a vector bundle Θ^+_k,n on Ê_T(pt). In the language of Section <ref>, Θ^+_k,n=p_*ℒ(N^Θ_k,n,0) is the direct image by the projection p E^(k)×Ê_T(pt)→Ê_T(pt) onto the second factor of the line bundle associated with the quadratic form

    N^Θ_k,n =2∑_i=1^kt_i(λ+(n-k)y)+∑_i=1^k∑_a=1^n(t_i-z_a)^2+k(k-1)y^2.

Here, as usual, the t_i are coordinates on the universal covering of E^k and z_a, y, λ are coordinates on the universal covering of Ê_T(pt). In fact only the terms involving t_i in N^Θ_k,n are determined by the transformation properties of the fibers Θ^+_k(z,y,λ). We choose the remaining terms to simplify the formulation of Theorem <ref> below.

The space of theta functions Θ̅^+_k(z,y,λ) satisfying the vanishing condition of Section <ref> is the generic fiber of a coherent subsheaf Θ̅^+_k,n of Θ^+_k,n on Ê_T(pt) (it is the intersection of kernels of morphisms ev_z_a,2 of coherent sheaves). The sheaves Θ̅^+_k,n and ℋ_T^ell(X_k,n) are closely related: there is a morphism

    φ Θ̅^+_k,n→ℋ_T^ell(X_k,n),

defined as follows. A section of Θ̅^+_k,n on an open set U is given by a function f(t;z,y,λ) on ℂ^k× U, which, as a function of t belongs to Θ^+_k(z,y,λ) and obeys the vanishing conditions

    f(z_a,z_a-y,t_3,…,t_k;z,y,λ)=0,      a=1,…,n.

The morphism φ sends f to (φ_If)_I⊂ [n],|I|=k where φ_If is the restriction of φ f to ι_I Ê_T(pt)≅Ê_T(pt):

    φ_If(z,y,λ)= f(t;z,y,λ)/∏_i≠ jθ(t_i-t_j+y)|_t=z_I,

cf. Section <ref>.
 Let D⊂Ê_T(pt) be the union of the hypertori z_a=z_b+y, 1≤ a≠ b≤ n and λ=jy, -n≤ j≤ n. The map φ f↦ (φ_If)_I⊂ [n], |I|=k given by formula (<ref>) is a well-defined injective morphism of 𝒪_Ê_T(pt)-modules
 
    φ Θ̅^+_k,n↪ℋ^ell_T(X_k,n),

 which is an isomorphism on Ê_T(pt)∖ D.


 We first prove that the morphism is well-defined. The function φ_If of z, y, λ defines a section of the line bundle ℒ(Q,0) with quadratic form Q=(N^Θ_k,n-∑_i≠ j(t_i-t_j+y)^2)|_t=z_I. An explicit calculation shows that
 
    Q=N_k,n|_t=z_I,  s=z_I̅.

 It follows that φ f is a meromorphic section of p_T*𝒯_k,n. By Lemma <ref> (which applies to any symmetric theta function obeying the vanishing condition), φ f is actually holomorphic.

To show that the morphism is injective, we use the fact that the weight functions ω_I^+ form a basis of Θ̅^+_k(z,y,λ) at the generic point of Ê_T(pt), see Proposition <ref>. Thus every local section of Θ̅^+_k,n can be written as linear combination of normalized weight functions with meromorphic coefficients. If this section is in the kernel of our morphism then its restriction to each component vanishes. By the triangularity property of weight functions of Lemma <ref> all coefficients must vanish and the kernel is trivial.

 We now construct the inverse map on the complement of D. A section s of ℋ^ell_T(X_k,n) on an open set U is a collection of sections s_I of 𝒯_k,n on the various components of p_T^-1(U) and agreeing on intersections. Then f=φ^-1s is
 
    f(t;z,y,λ)= ∑_I,Kψ_K(y,λ)^-1w_K^-(z_I,z,y,λ)s_I(z,y,λ)/∏_a∈ I, b∈I̅θ(z_a-z_b)θ(z_a-z_b+y)ω_K^+(t,z,y,λ).

 It is easy to check that this is a meromorphic section of Θ̅^+_k,n on U with poles at z_b=z_a+y, 1≤ a≠ b≤ n and at the zeros of ψ_K. It is regular at the apparent poles at z_a=z_b since the sections s_I agree on intersections of the components. Let us compute φ f:
 
    f(t;z,y,λ)/∏_i≠ jθ(t_i-t_j+y) =∑_I,Kw_K^-(z_I,z,y,λ)s_I(z,y,λ)/∏_a∈ I, b∈I̅θ(z_a-z_b)θ(z_a-z_b+y) w_K^+(t,z,y,λ).

 The orthogonality relations, see Corollary <ref>, imply
 
    φ_Jf =∑_I,Kw_K^-(z_I,z,y,λ) s_I(z,y,λ)/∏_a∈ I,  b∈I̅θ(z_a-z_b) θ(z_a-z_b+y)w_K^+(z_J,z,y,λ) =s_J(z,y,λ).*





 §.§ Symmetric group and G-equivariant cohomology classes


The symmetric group S_n on n letters acts on ℂ^n by permutation of coordinates. This action induces an action of S_n on the Grassmannians, their cotangent bundles X_k,n and on T so that the action map
T× X_k,n→ X_k,n is S_n-equivariant. The induced action on the cohomology schemes Ê_T(X_k,n) can be easily described: on Ê_T(pt)=E^n× E^2, S_n acts by permutations of the first n factors and σ∈ S_n sends the component ι_KÊ_T(pt) of Ê_T(X_k,n) to ι_σ(K)Ê_T(pt) so that the diagram

    [ Ê_T(pt)
     σ⟶Ê_T(pt);      ι_K↘  ↙ι_σ(K);         Ê_T(X_k,n) ]

commutes for any K⊂[n] with |K|=k elements. The structure map Ê_T(X_k,n)→Ê_T(pt) is S_n-equivariant and the quotient by the action of S_n is the G-equivariant elliptic cohomology scheme.

 The twisting bundle is S_n-equivariant, i.e., the S_n-action lifts to an S_n action on the bundle.

 This follows since the twisting bundle is the pull-back by an S_n-equivariant map of a bundle on E^(k)× E^(n-k) on which the action of the symmetric group is trivial.


In particular for each σ∈ S_n and admissible line bundle ℳ, we have an admissible line bundle σ^*ℳ and a map

    σ^* Γ_mer(Ê_T(X_k,n),ℳ)→Γ_mer(Ê_T(X_k,n),σ^*ℳ),

and also a map

    σ^* ℋ^ell_T(X_k,n)→σ^*ℋ^ell_T(X_k,n).

Let πÊ_T(pt)→Ê_G(pt)=Ê_T(pt)/S_n. Then we obtain an action of the symmetric group on π_*ℋ^ell_T(X_k,n).
 Let G=U(n)× U(1). The sheaf of G-equivariant elliptic cohomology classes is
 
    ℋ^ell_G(X_k,n)=π_*ℋ^ell_T(X_k,n)^S_n,

a coherent sheaf on Ê_G(pt)=E^(n)× E× E. Let ℒ∈Pic(Ê_G(pt)). The space of G-equivariant elliptic cohomology classes of degree ℒ on X_k,n is H^ell_G(X_k,n)_ℒ=Γ(Ê_G(pt),ℋ_G^ell(X_k,n)⊗ℒ). We set H^ell_G(X_n)_ℒ to be the 𝔥-module⊕_k=0^n H^ell_G(X_k,n)_ℒ, with the summandlabeled by k of 𝔥-weight -n+2k.




 §.§ Admissible difference operators

Recall that Ê_T(X_k,n) has a factor E× E corresponding to the U(1)-action on the cotangent spaces and the dynamical parameter. For j∈ℤ, let τ_j=τ_1^j be the automorphism of E× E such that

    τ_j(y,λ)=(y,λ+jy).
 Denote also by τ_j the automorphism id×τ_j of Ê_T(X_k,n)=E_A(Gr(k,n))× E× E. If ℒ is a line bundle on Ê_T(X_k,n) then τ_j lifts to a (tautological) bundle map ℒ→τ_j^*ℒ, also denoted by τ_j^*. It maps meromorphic sections to meromorphic sections and is thus a well-defined operator

    τ^*_j Γ_mer(Ê_T(X_k,n),ℒ)→Γ_mer(Ê_T(X_k,n),τ^*_jℒ).

 Let k=0,…,n and ℒ be a line bundle on Ê_T(pt), μ∈ 2ℤ, ν∈ℤ. An admissible difference operator on meromorphic sections of an
admissible line bundle ℳ_1=p_T^*ℒ_1⊗𝒯_k,n on Ê_T(X_k,n) of degree (ℒ,μ,ν) is a linear map φΓ_mer(Ê_T(X_k,n), ℳ_1)→Γ_mer(Ê_T(X_k+μ,n),ℳ_2)) such that
 =0pt
 
  (i) ℳ_2 is the admissible bundle p_T^*ℒ_2⊗𝒯_k+μ/2,n with ℒ_2=ℒ⊗τ_ν^*ℒ_1.
 
  (ii) For each section s of ℳ_1 and fixed point
 x_K∈ X_k+μ,n,
 
    ι_K^*φ(s)=∑_K'φ_K,K'τ_ν^*ι_K'^*s

 for some sections φ_K,K'∈Γ_mer(Ê_T(pt),ι^*_Kℳ_2⊗ι^*_K'τ^*_νℳ_1^-1).
 

By inserting the definition, we see that the line bundle of which φ_K,K' is a section is

    ι_K^*ℳ_2⊗τ_ν^*ι_K'^*ℳ_1^-1= ℒ⊗ι^*_K𝒯_k+μ/2,n⊗τ_ν^*ι_K'^*𝒯_k,n^-1.

This line bundle is independent of the admissible line bundle the operator acts on. It thus makes sense to let the same admissible difference operator act on sections of different admissible line bundles. We set

    𝒜_k,n(ℒ,μ,ν),      k=0,…,n,     0≤ k+μ≤ n,

to be the space of admissible difference operators of degree (ℒ,μ,ν).

It is convenient to extend the above definitions to the case of varying k. We denote by X_n=⊔_k=0^n X_k,n the disjoint union of cotangent bundles to Grassmannians of subspaces of all dimensions in ℂ^n. The extended elliptic cohomology scheme
is then

    Ê_T(X_n)=⊔_k=0^n Ê_T(X_k,n).

It comes with a map p_TÊ_T(X_n)→⊔_k=0^nÊ_T(pt).

 An admissible line bundle on Ê_T(X_n) is a line bundle whose restriction to each Ê_T(X_k,n) is admissible. Let ℒ=(ℒ^0,…,ℒ^n) be a line bundle on ⊔_k=0^nÊ_T(pt), μ,ν∈ℤ. An admissible difference operator of degree (ℒ,μ,ν) acting on sections of an admissible line bundle ℳ_1 is a linear map Γ_mer(Ê_T(X_n),ℳ_1)→Γ_mer(Ê_T(X_n),ℳ_2) restricting for each k=0,…,n such that k+μ∈{0,…,n} to an admissible difference operator
 
    Γ_mer(Ê_T(X_k,n),ℳ_1|_X_k,n)→Γ_mer(Ê_T(X_k+μ,n),ℳ_2|_X_k+μ,n),

 of degree (ℒ^k,μ,ν). We denote by
 
    𝒜_n(ℒ,μ,ν)=⊕_0≤ k,k+μ≤ n A_k,n(ℒ^k,μ,ν)

 the space of admissible difference operators of degree (ℒ,μ,ν).

 We will not need to consider operators on components for k such that k+μ∉{0,…,n}. However to have a correct definition we may set Ê_T(X_k,n) to be the empty set if k∉{0,…,n} and declare the space of sections of any line bundle on the empty set to be the zero vector space.




 §.§ Left and right moment maps


Examples of admissible difference operators are multiplication operators by sections of pull-backs of line bundles on Ê_T(pt). A subclass of these operators appear as coefficients in the defining relations of
the quantum group: they are the entries of the R-matrix and are functions of the dynamical and deformation parameter, and appear in the relations in two different guises: with and without “dynamical shift”. We borrow the terminology of <cit.>, where these two appearances are called the left and right moment maps.

Let ℒ be a line bundle on Ê_T(pt). We define two line bundles μ_ℓℒ, μ_rℒ on ⊔_k=0^nÊ_T(pt):
=0pt

  * μ_rℒ is the line bundle (ℒ,…,ℒ);

  * μ_ℓℒ is the line bundle (τ_n^*ℒ,τ_n-2^*ℒ,…,τ_-n^*ℒ).

 The left moment map is the map
 
    μ_ℓ Γ_mer(Ê_T(pt),ℒ)→𝒜_n(μ_ℓℒ,0,0),

 sending a section s to the operator whose restriction to 𝒜_k,n is the multiplication by τ_n-2k^*p_T^*s.

The right moment map is the map μ_rΓ_mer(Ê_T(pt),ℒ) →𝒜_n(μ_rℒ,0,0) sending s to the operator whose component in 𝒜_k,n is
the multiplication by p_T^*s.



 §.§ Sections of admissible bundles as a module over multiplication operators


Let ℒ∈Pic(Ê_T(pt)) and set 𝒜^0_n(ℒ)=(𝒜_n(ℒ,0,0)). Then the family 𝒜_n^0=(𝒜^0_n(ℒ))_ℒ∈Pic(Ê_T(pt)) is a commutative subalgebra graded by the Picard group of the base. It acts on meromorphic sections of admissible bundles by mapping Γ_mer(X_k,n,ℳ) to Γ_mer(X_k,n,ℳ⊗ p_T^*ℒ) for any admissible line bundle ℳ. Then the weight functions form a system of free generators of the module of sections of admissible line bundles over 𝒜^0_n in the following sense.
 Let ℒ∈Pic(Ê_T(pt)). Every section ω∈Γ_mer(Ê_T(X_k,n),p_T^*ℒ⊗𝒯_k,n) can be uniquely written as
 
    ω=∑_I⊂ [n], |I|=k a_I Stab(v_I),

 for some a_I∈𝒜^0_n(ℒ⊗ℒ_I^-1), where L_I is the line bundle of Proposition <ref>.

 Denote by Y_I=ι_IÊ_T(pt) the component labeled by I. Suppose that ω is a meromorphic section vanishing on Y_J for all J>I and such that ω|_Y_I≠ 0. By Lemma <ref>, we can subtract from ω a multiple of c ^*w_I^+ to get a section that vanishes on Y_J, J≥ I. By induction we may subtract from ω a suitable linear combination of weight functions to get 0.




 §.§ S_n-equivariant admissible difference operators

 An admissible difference operator is called S_n-equivariant if it commutes with the action of the symmetric group on sections.


 Let ℒ be an S_n-equivariant line bundle on Ê_T(pt). An admissible difference operator φ of degree (ℒ,μ,ν) is S_n-equivariant if and only if its matrix elements φ_K,K' obey
 
    σ^*φ_σ(K),σ(K')=φ_K,K'.





 §.§ Graded algebras, graded modules

Let Q be a group. Recall that an Q-graded algebra A over ℂ is a collection (A_γ)_γ∈ Q of complex vector spaces with associative linear multiplication maps A_γ⊗ A_γ'→𝒜_γγ', a⊗ b↦ a· b. Let P be a set with a left action of Q. A P-graded (left) module over A is a collection (M_p)_p∈ P of complex vector spaces indexed by P together with linear action maps A_γ⊗ M_p→ A_γ· p, a⊗ m↦ a· m, obeying (a· b)· m=a·(b· m). A unital Q-graded algebra is an Q-graded algebra with an identity element 1∈ A_e in the component indexed by the identity element e of Q. We require 1 to act as the identity on P-graded modules.



 §.§ The grading of admissible difference operators

Let Q=Pic(Ê_T(pt))⋉ 2ℤ×ℤ be the product of the Picard group of Ê_T(pt)≅ E^n+2 by 2ℤ×ℤ with group law

    (ℒ,μ,ν)(ℒ',μ',ν') =(ℒ⊗τ^*_νℒ', μ+μ',ν+ν').

 The collection (𝒜_n(ℒ,μ,ν))_(ℒ,μ,ν)∈ Q with the composition of operators is a unital Q-graded algebra.

 An admissible difference operator of degree (ℒ',μ',ν') sends a section of an admissible line bundle ℳ_1=p_T^*ℒ_1⊗𝒯_k,n to a section of ℳ_2=p_T^*ℒ_2⊗𝒯_k+μ'/2,n with ℒ_2=ℒ'⊗τ^*_ν'ℒ_1. An operator of degree (ℒ,μ,ν) sends this section to a section of p_T^*ℒ_3⊗ T_k+μ/2+μ'/2,n with
 
    ℒ_3=ℒ⊗τ^*_νℒ_2 =ℒ⊗τ_ν^*ℒ'⊗τ^*_ν+ν'ℒ_1.

It is clear that the μ-components of the degree add, so the composition has degree (ℒ⊗τ_ν^*ℒ',μ+μ', ν+ν'), as required. The identity element is the multiplication by constant function 1, a section of the trivial bundle 𝒪.


 There is a slight abuse of notation, since 𝒜(ℒ,μ,ν) is defined for a line bundle ℒ and not for its equivalence class. The point is that 𝒜(ℒ,μ,ν) for equivalent bundles ℒ are canonically isomorphic: if φ is an admissible difference operator of degree (ℒ,μ,ν) and ψℒ→ℒ' is an isomorphism then φ'=ψ∘φ∘ψ^-1 is an difference operator of degree (ℒ',μ,ν). This establishes the isomorphism
 
    ψ̅ 𝒜(ℒ,μ,ν)→𝒜(ℒ',μ,ν),

 which we claim is independent of ψ. Indeed any two choices of ψ differ by the composition with an automorphism of ℒ. Since Aut(ℒ)=ℂ^×, ψ and ψ' differ by multiplication by a nonzero scalar which does not affect ψ̅.

Let P be the set of pairs (ℒ,μ) with ℒ∈Pic(Ê_T(X_n)) and μ∈ℤ. Then Q acts on P via

    (ℒ,μ,ν)·(ℒ',μ') = (ℒ⊗τ_ν^*ℒ',μ+μ').

Let ℳ=p_T^*ℒ_1⊗𝒯_k,n be an admissible bundle on X_k,n.

Admissible difference operators map sections of admissible line bundles to sections of admissible line bundles. This is formalized as follows.
 The collection of vector spaces Γ_mer(Ê_T(X_k,n),p_T^*ℒ⊗ T_k,n), labeled by (ℒ,μ), with μ=-n+2k is a P-graded module over the Q-graded unital algebra 𝒜_n of admissible difference operators.

 This is an immediate consequence of the definition, see Definition <ref>(i).




§ ACTION OF THE ELLIPTIC DYNAMICAL QUANTUM GROUP

In this section we construct an action of the elliptic dynamical quantum group associated with 𝔤𝔩_2 on the extended equivariant elliptic cohomology Ê_T(X_n) of the union of cotangent bundles of the
Grassmannians of planes in ℂ^n. The action is by S_n-equivariant admissible difference operators acting on admissible line bundles on the cohomology scheme. Thus each generator L_ij(w)  of the elliptic dynamical quantum group acts on sections of any admissible line bundle by an admissible difference operator of some degree (ℒ_ij(w),μ_ij,ν_ij) which we give below. We also compute the action on T-equivariant elliptic cohomology classes and use the S_n-equivariance to show that the action descends to an action on G-equivariant classes, with G=U(n)× U(1).

We construct the action in such a way that at the generic fibre of Ê_T(X_k,n)→ E_T(pt)=E^n× E (i.e., for fixed z_1,…,z_n,y) the map (<ref>) defines a morphism of representations from the tensor product of evaluation representations. In other words, suppose that

    L_ij(w)v_I=∑_K L_ij(w,z,y,λ)_I^Kv_K,

for some meromorphic coefficients L_ij(w,z,y,λ)_I^K. Then we want that

    L_ij(w)Stab(v_I) =∑_K L_ij(w,z,y,λ)_I^KStab(v_K).

The matrix coefficients L_ij(w,z,y,λ)_I^K are certain meromorphic functions of z,y,λ with theta function-like transformation properties and can thus be considered as meromorphic sections line bundles on Ê_T(pt). Therefore each summand on the right-hand side is a meromorphic section of an admissible line bundle. The content of the following theorem is that the sum defines uniquely an admissible difference operator.
  =0pt

  (i) The formula (<ref>) uniquely defines admissible difference operators L_ij(w), i,j∈{1,2}, of degree (ℒ_ij(w),2(i-j),ϵ(j)) with ϵ(1)=-1, ϵ(2)=1, for some S_n-equivariant line bundle ℒ_ij(w) on Ê_T(pt).

  (ii) These operators obey the RLL relations of the elliptic dynamical quantum group in the form
 
    μ_ℓ R(w_1-w_2,y,λ)^(12) L(w_1)^(13) L(w_2)^(23) = L(w_2)^(23)L(w_1)^(13)μ_rR(w_1-w_2,y,λ)^(12).

 Here the coefficients of the quadratic relations are in 𝒜^0_n and the action of μ_ℓ, μ_r is on each matrix element of R.



The proof of this theorem is by explicit description of the action and is parallel to the case of Yangians and affine quantum enveloping algebras, <cit.>. We give the formulae for the action in Sections <ref>, <ref> and <ref>. The proof of Theorem <ref> is in Section <ref>.

By Theorem <ref>, the generators L_ij(w) send meromorphic sections of admissible bundles to meromorphic sections of admissible bundles. The next result gives a more precise control on the poles of coefficients. We give the action on holomorphic sections, i.e., equivariant elliptic cohomology classes, both for the torus T=U(1)^n× U(1) and the group G=U(n)× U(1).

  =0pt

  (i) Let D be the divisor on Ê_T(pt) whose components are the hypersurfaces defined by equations z_a+y=w, for 1≤ a≤ n and λ+yj=0, for j=-n…,n-1,n. Then L_ij(w) maps H^ell_T(X_n)_ℒ to H^ell_T(X_n)_τ_ϵ(j)^*ℒ⊗ℒ_ij(w)(D) for any ℒ∈Pic(Ê_T(pt)).

  (ii) Let H^ℓ_G(X_n)_ℒ be the space of G-equivariant elliptic cohomology classes of degree ℒ for G=U(n)× U(1), see Definition <ref>. Let πÊ_T(pt)→Ê_G(pt) be the canonical projection. View line bundles on Ê_G(pt) as S_n-equivariant line bundles on Ê_T(pt). Then the opera­tors L_ij(w) induce well-defined operators from H^ell_G(X_n)_ℒ to H^ell_G(X_n)_τ_ϵ(j)^*ℒ⊗ℒ_ij(w)(π(D)) for each ℒ∈Pic(Ê_G(pt)).


The proof of this theorem is contained in Section <ref>.
 Let qÊ_T(pt)=E_T(pt)× E→ E_T(pt) be the projection onto the first factor. Since the action of the generators L_ij(w) is by admissible difference operators it preserves the fiber of q_*ℋ_T^ell(X_n) at a generic point of the non-extended E_T(pt). If we realize this fiber as a certain space of functions of λ and tensor with all meromorphic functions of λ we get a representation of the quantum group in the sense of Section <ref>. By construction, it is isomorphic to the tensor product of evaluation representations. Thus we can think of the action of the quantum group on equivariant elliptic cohomology classes as a tensor product of evaluation representations with variable evaluation points and deformation parameter.




 §.§ Action of the Gelfand–Zetlin subalgebra

The Gelfand–Zetlin subalgebra is the commutative subalgebra generated by L_22(w) and the determinant Δ(w). As shown in Section <ref> these operators act diagonally in the basis ξ_I of V(z_1)⊗⋯⊗ V(z_n). It follows that the vectors

    ξ̂_I=∑_|J|=kw^-_J(z_I,z,y,λ)/∏_a∈ I, b∈I̅θ(z_a-z_b+y)Stab(v_J),

(cf. (<ref>)), which by construction are sums of sections of certain admissible line bundles, are eigenvectors of the Gelfand–Zetlin subalgebra. It turns out that they are sections of admissible bundles with support on a single irreducible component of Ê_T(X_n):
 Let I,K⊂ [n], |I|=|K|=k. The restriction of ξ̂_I to the component of Ê(X_k,n) labeled by K is

    ι_K^*ξ̂_I= ∏_a∈ I, b∈I̅θ(z_a-z_b),   I=K,
    
    0,    otherwise.


 From the definition of ξ̂_I and Stab(v_J) we have

    ι_K^*ξ̂_I =ι_K^*(∑_J  w_J^-(z_I,z,y,λ)/∏_a∈ I,  b∈I̅θ(z_a-z_b+y) w^+_J(t,z,y,λ) ) 
     =1/∏_a∈ I,  b∈I̅θ(z_a-z_b+y)∑_J w_J^-(z_I,z,y,λ) w^+_J(z_K,z,y,λ),

which, using Corollary <ref>, proves our statement.


Thus we can write any section (s_I)_I⊂[n] of an admissible bundle as linear combination

    ∑s_Iξ̂_I/∏_a∈ I, b∈I̅θ(z_a-z_b).

Since the ξ̂_I are eigenvectors the action of the Gelfand–Zetlin algebra is given by admissible difference operator with diagonal matrices of coefficients.

The action of the determinant Δ(w) is easiest to describe: it is given by an admissible difference operator of degree (ℒ_Δ(w),0,0), where ℒ_Δ(w)=ℒ(N_Δ,v_Δ(w)) is the line bundle associated with the data

    N_Δ(z,y,λ)= ∑_a=1^n (2z_a+y)y,      v_Δ(w;z,y,λ)=-wy.

Since N_Δ and v_Δ are symmetric under permutations of the variables z_i, the corresponding bundle is naturally S_n-equivariant. The determinant acts on sections of any admissible line bundle ℒ_1 on Ê_T(X_k,n) it acts by multiplication by the section

    ∏_i=1^nθ(w-z_i+y)/θ(w-z_i)

of ℒ_Δ(w).

The action of L_22(w) is by an operator of degree (ℒ_22(w),0,1). It is defined on the components by

    ι^*_KL_22(w)s=∏_i∈ Kθ(w-z_i)/θ(w-z_i-y)τ_1^*ι_K^*s,

and ℒ^k_22(w)=ℒ(2k∑_a=1^nz_ay,kwy), which is an S_n-equivariant line bundle.
 These formulae define S_n-equivariant admissible difference operators Δ(w)∈𝒜(ℒ_Δ(w),0,0) and L_22(w)∈𝒜(ℒ_22(w),0,1).

 Both operators are defined by diagonal matrices (φ_K,K') in the notation of Definition <ref>. It is straightforward to check that the diagonal matrix elements φ_K,K are sections of the correct line bundle. The equivariance property of Lemma <ref> is clearly satisfied. Moreover the divisor of poles does not contain any diagonal Δ_I={z∈ E^n+2 z_i=z_j, ∀  i,j∈ I}, I⊂[n] so that the difference operator maps meromorphic sections to meromorophic sections. It remains to check that the sections on the different components coincide on their intersections, namely that for every a∈ K, b∈K̅,
 
    φ_K,K|_z_a=z_b=φ_K̃,K̃|_z_a=z_b,     K̃=K∖{a}∪{b}.

 This can be checked directly but also follows from the equivariance condition for the permutation of a and b.




 §.§ Action of L_12 and L_21

Let k=1,…,n and K⊂[n] with |K|=k-1, then

    ι^*_K L_12(w)s =(-1)^kθ(y)∑_a∈K̅θ(λ+w-z_a+(n-2k+1)y)/θ(w-z_a-y)
    ×∏_j∈ Kθ(w-z_j)/θ(w-z_j-y)∏_j∈ Kθ(z_a-z_j-y) /∏_j∈K̅∖{a}θ(z_a-z_j) ι_K∪{a}^*τ_1^*s.

 L_12(w) is an S_n-equivariant admissible difference operator of degree (ℒ_12(w),-2,1) with
 
    ℒ^k_12(w)=ℒ(-(λ+(n-2k)y)^2,-w(λ+(n-k+1)y)).


Let k=0,…,n-1 and K⊂[n] with |K|=k+1, then

    ι^*_K L_21(w)s =(-1)^n-kθ(y)/θ(λ)θ(λ-y)∑_a∈ Kθ(λ-w+z_a)/θ(w-z_a-y)
    ×∏_j∈ K∖{a}θ(w-z_j)/θ(w-z_j-y)∏_j∈K̅θ(z_j-z_a-y) /∏_j∈ K∖{a}θ(z_j-z_a) ι_K∖{a}^*τ_-1^*s

 L_21(w) is an S_n-equivariant admissible difference operator of degree (ℒ_21(w),2,-1) with
 
    ℒ^k_21(w)=ℒ(-λ^2-(n-2k+2)y^2+2y(λ-∑_i=1^nz_i),w(λ-(k+1)y)).



Lemmas <ref> and <ref> are proved the same way as Lemma <ref>. The only new feature is the appearance of simple poles on diagonals z_i=z_j and it is thus not a priori clear that these operators map meromorphic sections to meromorphic sections in the sense of Definition <ref>. The point is that when acting on meromorphic sections, these poles cancel by the equivariance conditions. For example let us consider the behaviour of φ=L_12(w)s in the vicinity of the diagonal z_a=z_b. The matrix element φ_K,K∪{a} has a simple pole there if b∈K̅∖{a} and so has the matrix element φ_K,K'∪{b} which by equivariance is obtained from φ_K,K∪{a} by the transposition σ_12 of a, b. In local coordinates and trivializations compatible with the S_n-action, by setting f_j=ι_K∪{j}τ^*_1s, j=a,b, the potentially singular term in ι_K^*L_12(w)s at z_a=z_b has the form

    g(z,y,λ)/z_a-z_bf_a(z,y,λ) +g(s_abz,y,λ)/z_b-z_af_b(σ_12z,y,λ).

Since s_abz=z and f_a=f_b on the diagonal z_a=z_b, the poles cancel. The same argument works for L_21.



 §.§ Action of L_11

Since L_22(w) is an invertible admissible difference operator, the action of L_11(w) can be obtain from the action of the Gelfand–Zetlin algebra and the action of L_12, L_21 via the formula for the determinant

    Δ(w)=μ_ℓ(θ(λ))/μ_r(θ(λ)) (L_11(w+y)L_22(w)-L_21(w+y)L_12(w)).
 Here θ(λ) is considered as a section of the bundle ℒ(N,0) on E^2 with N(y,λ)=λ^2.



 §.§ Proof of Theorems <ref> and <ref>

Theorem <ref>(i) for L_22, L_12 and L_21 follows from Lemma <ref>, Lemma <ref> and Lemma <ref>, respectively. The operator L_11 can be expressed as composition of these admissible difference operators via the determinant and is thus also admissible. Part (ii) follows by construction. To prove Theorem <ref>(i) we need to check that in the matrix elements of L_ij(w) only simple poles at z_i=w+y and λ=jy, j∈ℤ appear. This is clear from the explicit formulae except for L_11(w). To prove it for this operator, we use two formulae for it: one is using the definition and the orthogonality relations, and the other expressing it in terms of the other L_ij and the determinant, as in Section <ref>.

The first formula gives

    ι_L^*L_11(w)s=∑_I,J,Kw^-_K(z_I)L_11(w)_K^Jw^+_J(z_L)/∏_a∈ I, b∈I̅θ(z_a-z_b) θ(z_a-z_b+y)τ_-1^*ι_I^*s.

The matrix elements L_11(w)_K^J of L_11(w) in the tensor basis v_I of (ℂ^2)^⊗ n are sums of products of matrix elements of R-matrices and have at most simple poles at z_a=w+y and possible poles at λ=jy, j∈ℤ, see (<ref>). Thus the right-hand side of the (<ref>) has (among other poles) at most simple poles at z_a=w+y. The second formula is in terms of the determinant:

    L_11(w)s=(θ(λ)/θ(λ-yμ)Δ(w-y) +L_21(w-y)L_12(w))L_22(w+y)^-1s,

for s∈ H_T^ell(X_k,n)_ℳ with μ=-n+2k. From this formula and the explicit expression of L_12, L_21, L_22 we see that only simple poles at λ=yμ, -n≤μ≤ n occur and that the remaining apparent poles at z_a-z_b=0, z_a-z_b+y=0 in (<ref>) are spurious.

Finally Theorem <ref>(ii) holds since the bundles ℒ_ij(w) are S_n-equivariant (and can thus be viewed as line bundles on the quotient) and the action is given by S_n-equivariant difference operators.



§ SHUFFLE PRODUCTS AND STABLE ENVELOPES FOR SUBGROUPS

The stable envelope of <cit.> is a map from the equivariant cohomology of the fixed point set for a torus action on a Nakajima variety to the equivariant cohomology of the variety. The goal of this section is to extend this interpretation of the stable envelope to the elliptic case for cotangent bundles of Grassmannians. In our construction the stable envelope is built out of weight functions, which in turn are obtained from shuffle products of elementary weight functions associated with the one-point spaces T^*Gr(0,1), T^*Gr(1,1). Thus the first step is to extend the fiber-by-fiber construction of the shuffle product of Section <ref> to a shuffle product defined on sections of the coherent sheaf Θ̅^+_k,n on Ê_T(pt). By using the isomorphism (outside the divisor D) of Theorem <ref>, we get a shuffle product on the sections of the sheaf of elliptic cohomology classes ℋ^ell_T(X_k,n). The n-fold shuffle product of classes in ℋ^ell_T(X_k,1), k=0,1 is then essentially the stable envelope.

We propose to view shuffle products of factors of an arbitrary number of factors as stable envelopes corresponding to subgroups of T. Their geometric interpretation is that they correspond to maps from the cohomology of the fixed point set for the action of a subgroup of the torus T, cf. <cit.>.

The basic case, which as we shall see corresponds to the shuffle product of two factors, is the subgroup B_m⊂ U(1)^n

    B=B_m={(z,…,z_m,1,…,1)∈ A  z∈ U(1)},

isomorphic to U(1).

Fixed points for the action of this subgroup on Gr(k,n) are k-planes of the form V_1⊕ V_2, with V_1 in the span of the first m coordinate axes and V_2 in the span of the last n-m coordinate axes. Thus the fixed point set decomposes into connected components according to the dimension of V_1. Each of these components is a product of Grassmannians. Similarly, the B_m-invariant part of the cotangent space at a fixed point splits as a direct sum of cotangent spaces at the factors and we get an isomorphism

    X_k,n^B_m≅⊔_d=0^k X_d,m× X_k-d,n-m.

As above we consider the action of A_n=U(1)^n on X_k,n. Then the embedding X_d,m× X_k-d,n-m↪ X_k,n is A_m× A_n-m= A_n-equivariant. The Künneth formula <cit.> predicts that this embedding induces a map

    E_A_m(X_d,m)× E_A_n-m(X_k-d,n-m)→ E_A_n(X_k,n),

In the description as a fiber product,

    E_A_m(X_d,m)=E^(d)× E^(m-d)×_E^(m)E^m,

and the map is the obvious one: ((t',s',z'),(t”,s”,z”))↦ (p(t',t”),p(s',s”),z',z”). Here t'∈ E^(d), t”∈ E^(k-d), p E^(d)× E^(k-d)→ E^(k) is the canonical projection and similarly for the other factors.

As in Section <ref>, we consider the extended equivariant elliptic cohomology Ê_T_n(X_k,n)=E_A_n(X_k,n)× E^2 for the torus T_n=A_n× U(1) where the additional U(1) factor acts by multiplication on each cotangent space. We then have the corresponding embedding

    Ê_T_m(X_d,m)×_E^2Ê_T_n-m(X_k-d,n-m)→Ê_T_n(X_k,n),

where the map to E^2 is the projection onto the second factor. Both are schemes over Ê_T_n(pt)=Ê_T_m(pt)×_E^2E_T_n-m(pt).

 The shuffle product of Proposition <ref> defines a map

    * τ^*_n”-2k”Θ̅^+_k',n'⊠Θ̅^+_k”,n”→Θ̅^+_k,n⊗ℒ_k',k”,n',n”

of sheaves of 𝒪_Ê_T(pt)-modules, where k=k'+k”, n=n'+n” and ℒ_k',k”,n',n”= ℒ(k”y((n'-k')y-2∑_a=1^n'z_a),0)∈Pic(Ê_T_n(pt)).

 The sheaf Θ^+_k,n is defined by the quadratic form N^Θ_k,n(t,z,y,λ), see (<ref>). Let us write t=(t',t”), z=(z',z”), with t'=(t_1,…,t_k'), t”=(t_k'+1,…,t_k) and similarly for z. Then τ^*_n”-2k”Θ̅^+_k',n'⊠Θ̅^+_k”,n” is associated with the quadratic form
 
    M(t,z,y,λ)=N^Θ_k',n'(t',z',y,λ+y(n”-2k”)) +N^Θ_k”,n”(t”,z”,y,λ).

 The shuffle product maps a section of this bundle to a section of a bundle associated with the sum of this quadratic form and the quadratic forms of the theta functions in φ^+, see Proposition <ref>, namely
 
    M(t,z,y,λ) + ∑_j=1^k'∑_l=k'+1^k ((t_j-t_l+y)^2-(t_j-t_l)^2)
     +∑_l=k'+1^k ∑_a=1^n'(t_l-z_a+y)^2+∑_j=1^k'∑_b=n'+1^n (t_j-z_b)^2.

 It is straightforward to verify that this is equal to
 
    N^Θ_k,n(t,z,y,λ)+k”y((n'-k')y-2∑_a=1^n'z_a).

 This shows that the shuffle product takes values in Θ^+_k,n⊗ℒ_k',k”,n',n”. The fact that it actually takes values in the subsheaf defined by the vanishing condition follows from Proposition <ref>.


 Let k=k'+k”, n=n'+n”. The unnormalized stable envelope associated with the component X_k',n'× X_k”,n” of the fixed point set X_k,n^B_n' is the shuffle product map
 
    Stab τ_n”-2k”^*Θ̅^+_k',n'⊠Θ̅^+_k”,n”→Θ̅^+_k,n⊗ℒ_k',k”,n',n”

 of sheaves of 𝒪_Ê_T_n(pt)-modules.

By using the isomorphism Θ̅_k,n≅ℋ_T_n^ell(X_k,n) on the complement of the divisor D of Theorem <ref>, we obtain a map

    Stab τ_n”-2k”^*ℋ^ell_T_n'(X_k',n') ⊠ℋ^ell_T_n”(X_k”,n”) →ℋ^ell_T_n(X_k,n)⊗ℒ_k',k”,n',n”,

on Ê_T(pt)∖ D.

More generally, we may consider subgroups B=B_n_1×⋯× B_n_r⊂ U(1)^n whose fixed point sets have components X_k_1,n_1×⋯× X_k_r,n_r and define stable envelopes given by r-fold shuffle products and thus by compositions of stable envelopes for two factors.

Two special cases give the stable envelope of Section <ref> and the action of the elliptic dynamical quantum group.

In the first case we take B=U(1)^n. The fixed points are isolated and labeled by I⊂[n]. We think of the fixed point labeled by I as a product X_k_1,1×⋯× X_k_n,1 with k_i=1 if i∈ I and k_i=0 otherwise. The unnormalized stable envelope on the component labeled by I⊂ [n] is then

    Stab ⊠_j=1^n τ^*_w(j,I)ℋ^ell_T_1(X_k_i,1) →ℋ^ell_T_1(X_k_i,1)⊗ℳ_I,

(the factors are ordered from left to right) for some suitable line bundle ℳ_I∈Pic(Ê_T_n(pt)) obtained as tensor product of line bundles L_k',k”,n',n”. In this case the map is defined everywhere, not just on the complement of D, since Θ̅_k,1≅ℋ^ell_T_1(X_k,1) on Ê_T_1(pt).

The stable envelope of Section <ref> is obtained by taking the tensor product with suitable line bundles τ^*_w(j,I)ℳ_k_j on Ê_T_n(pt) so that ⊕_k=0^1Γ(X_k,1,ℋ^ell_T_1(X_k,1)⊗ℳ_k) is identified with ℂ^2 via the basis ω_0^+,ω_1^+, passing to global sections and normalizing by dividing by ψ_I.

In the second case we reproduce the construction of Section <ref> in the global setting. Thus we consider the stable envelope for two factors X_d,1× X_k-d,n⊂ X_k,n+1.
We obtain two maps

    ⊕_d=0^1τ^*_n-2(k-d)Θ̅^+_d,1⊠Θ̅^+_k-d,n⊗ℒ_d,k-d,1,n^-1→Θ̅^+_k,n+1
        ⊕_d=0^1τ^*_1-2dΘ̅^+_k-d,n⊠Θ̅^+_d,1⊗ℒ_k-d,d,n,1^-1→Θ̅^+_k,n+1

which are invertible at a generic point. Since Θ̅^+_k,1 is isomorphic to ℋ^ell_T_1(X_k,1) we get a map

    ⊕_d=0^1τ^*_1-2dΘ̅^+_k-d,n⊠ℋ^ell_T_1(X_d,1) ⊗ℒ_k-d,d,n,1^-1
        →⊕_d=0^1τ^*_n-2(k-d)ℋ^ell_T_1(X_d,1)⊠Θ̅^+_k-d,n⊗ℒ_d,k-d,1,n^-1,

defined on some dense open set. This map contains the information of the action of the elliptic dynamical quantum group on the elliptic cohomology of Ê_T_n(X_n). The action of the generators is given as explained in Section <ref>: one needs as above to take the tensor product with a suitable line bundle to associate elliptic cohomology classes ω^+_1,ω^+_0 with the standard basis of ℂ^2. Then we are in the setting of Section <ref> and we obtain an action of the elliptic dynamical quantum group which is up to gauge transformation the one described in the previous section.



§ AXIOMATIC DEFINITION OF ELLIPTIC STABLE ENVELOPES


In this section we give an axiomatic definition of the elliptic stable envelopes in the spirit of Maulik–Okounkov <cit.>, see also <cit.>.

Recall that c ^*w^+_I is a meromorphic section (with controlled denominators) of an appropriate line bundle over Ê_T(X_k,n). The scheme Ê_T(X_k,n) has components Y_J=ι_J Ê_T(pt), and the restriction of a section to Y_J is the result of substituting the variables t_i by z_J=(z_i)_i∈ J.

A meromorphic section of an admissible line bundle p_T^*ℒ(N,0)⊗𝒯_k,n restricted to Y_J can be written as a meromorphic function F^n+2→ whose transformation properties with respect to the lattice ^n+2+τ^n+2 are determined by p_T^*ℒ(N,0)⊗𝒯_k,n, see Remark <ref>. Below we will consider special forms of such functions.

 For any I the section c ^*w^+_I satisfies the following properties.
=0pt

  * It is a meromorphic section of an admissible line bundle p_T^*ℒ(N,0)⊗𝒯_k,n.

  * The restriction of c ^*w^+_I to Y_I, written as a function ^n+2→ with transformation pro­per­ties determined by p_T^*ℒ(N,0)⊗𝒯_k,n, is
 
    ∏_a∈ I,  b∈I̅θ(z_a-z_b+ϵ(a,b)y)/∏_a∈ Iθ(λ-(w(a,I)+1)y),

where ϵ(a,b) is defined in Lemma <ref> and w(a,I) is defined in (<ref>).

  * The restriction of c ^*w^+_I to any Y_J, written as a function ^n+2→ with transformation properties determined by p_T^*ℒ(N,0)⊗𝒯_k,n, is of the form

    1/ψ_I∏_a∈ J∏_b∈J̅,  b<aθ(z_a-z_b+y) · F_I,J,

where F_I,J is a holomorphic function.

Moreover, these three properties uniquely determine c ^*w^+_I.


 From the second property one can calculate the quadratic form

    N_I =-2∑_a∈I̅n(a,I)z_ay-2∑_a∈ Iz_a(λ+n(a,I̅)y)
     +(k(n-k)-∑_a∈ In(a,I̅))y^2-∑_a∈ I(λ- (n(a,I)+1)y+n(a,I̅)y)^2,

cf. (<ref>).


Lemma <ref> (ii) implies the triangularity property
=0pt

  * the restriction of c ^*w^+_I to Y_J is 0 unless J≤ I.

According to Theorem <ref> this property is a consequence of the three properties listed.


The third listed property is a local version of a support condition used in the axiomatic description of cohomological stable envelopes in <cit.>; see also the corresponding axiom in K-theory in <cit.>.


The first two properties of c ^*w^+_I are claimed in Proposition <ref> and Lem­ma <ref>(ii). Inspecting the explicit formula for w^+_I in Section <ref> one finds that, after substitution t_i by z_a, a∈ J, all non-zero terms are of the form (<ref>), which proves the third property.

Now we prove that the three properties uniquely determine c ^*w_I^+. Let a section satisfy the three listed properties, and let κ_I be the difference of that section and c ^*w_I^+.
Assume that κ_I is not 0. Then there exists a J such that κ_I restricted to Y_J is not 0. For a total ordering ≺ refining the partial order < on the cardinality k subsets of [n] let us choose J to be the largest with the property κ_I|_Y_J≠0. We have J≠I because of the second property.

We claim that κ_I|_Y_J, written as a function ^n+2→ with transformation properties determined by p_T^*ℒ(N_I,0)⊗𝒯_k,n, is of the form

    1/ψ_I∏_a∈ J∏_b∈J̅,  b<aθ(z_a-z_b+y) ·∏_a∈ J∏_b∈J̅, b>aθ(z_b-z_a)· F_1,

where F_1 is holomorphic. The fact that this function can be written in the form

    1/ψ_I∏_a∈ J∏_b∈J̅, b<aθ(z_a-z_b+y) · F_2,

with F_2 holomorphic, is explicit from the third property. We need to prove that F_2 is the product of ∏_a∈ J∏_b∈J̅,  b>aθ(z_b-z_a) and a holomorphic function. Let a∈ J, b∈J̅ and b>a. Denote J'=(J-{a})∪{b}. Observe that J<J' and hence J≺ J'. From the choice of J therefore it follows that κ_I restricted to Y_J' is 0. The diagonal Δ_J,J'={z_a=z_b} is included both in Y_J and Y_J', hence we obtain that the substitution of z_a=z_b into κ_I|_Y_J vanishes. It follows that the function F_2 vanishes on the hyperplane z_a=z_b and its translates by the lattice ^n+2+τ^n+2. The zeros of θ(z_b-z_a) are exactly these hyperplanes and are of first order, therefore F_2 can be written as a product of θ(z_b-z_a) times a holomorphic function. Iterating this argument for all (a,b) with a∈ J, b∈J̅, b>a we obtain that (<ref>) is in fact of the form (<ref>), what we claimed.

Observe that the product of theta functions in (<ref>) is the numerator of c ^*w_J^+|_Y_J. Hence we obtain that (<ref>) further equals

    ∏_a∈ Jθ(λ-(w(a,J)+1)y)/ψ_I· c ^*w_J^+|_Y_J· F_1.

Since the transformation properties of κ_I|_Y_J are determined by p_T^*ℒ(N_I,0)⊗𝒯_k,n, and those of c ^* w_J^+|_Y_J are determined by p_T^*ℒ(N_J,0)⊗𝒯_k,n, we have that the transformation properties of

    ∏_a∈ Jθ(λ-(w(a,J)+1)y)/ψ_I· F_1

are determined by p_T^*ℒ(N_I-N_J,0) – the factor 𝒯_k,n canceled.

Let a∈ J ∩I̅, and consider (<ref>) as a function of z_a, let us call it f(z_a). Since the first factor (the fraction) only depends on λ and y, f is a holomorphic function of z_a for generic y,λ. Comparing the z_a dependence of N_I and N_J we obtain that

    f(z_a+τ)= e^-2π i (λ+sy) f(z_a),      f(z_a+1)=f(z_a),

for some integer s. Using the 1-periodicity, we expand

    f(z_a)=∑_m∈ℤ a_m  e^2π i m z_a,

and using the first transformation property of (<ref>) we obtain

    ∑_m a_m  e^2π i m z_a(  e^2π i m τ -  e^-2π i (λ+sy))=0,

implying a_m=0 for all m∈ℤ. We obtained F_1=0, and in turn, κ_I|_Y_J=0. This is a contradiction proving that κ_I is 0 on all Y_J.




 §.§ Acknowledgments
 We thank Nora Ganter and Mikhail Kapranov for explanations on equivariant elliptic co­ho­mo­logy. G.F. was supported in part by the National Centre of Competence in Research “SwissMAP – The Mathematics of Physics” of the Swiss National Science Foundation. R.R. was supported by the Simons Foundation grant #523882. A.V. was supported in part by NSF grant DMS-1362924 and Simons Foundation grant #336826. We thank the Forschungsinstitut für Mathematik at ETH Zürich and the Max Planck Institut für Mathematik, Bonn, where part of this work was done, for hospitality.

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