HOW TO REGULARIZE SINGULAR VECTORS AND KILL THE DYNAMICAL WEYL GROUP

K. STYRKAS^{⋆}, V. TARASOV^{⋄,1}, AND A. VARCHENKO^{⋆,2}

⋄St.Petersburg Branch of Steklov Mathematical Institute Fontanka 27, St.Petersburg 191011, Russia

⋆Department of Mathematics, University of North Carolina at Chapel Hill Chapel Hill, NC 27599-3250, USA

Abstract. Let 𝔤 be a simple Lie algebra, and let M_{λ} be the Verma module over 𝔤 with highest weight λ. For a finite-dimensional 𝔤-module U we introduce a notion of a regularizing operator, acting in U, which makes the meromorphic family of intertwining operators Φ : M_{λ+μ} → M_{λ} ⊗ U holomorphic, and conjugates the dynamical Weyl group operators A_{w}(λ) ∈ End(U) to constant operators. We establish fundamental properties of regularizing operators, including uniqueness, and prove the existence of a regularizing operator in the case 𝔤 = 𝔰𝔩_{3}.

1. Introduction

Let 𝔤 be a simple Lie algebra. Let M_{λ} be the Verma module over 𝔤 with highest weight λ, and 1_{λ} ∈ M_{λ} the highest weight vector. A fundamental object of the representation theory of 𝔤 is a family of intertwining operators

Φ_{λ}^{u} : M_{λ+μ} → M_{λ} ⊗ U, 1_{λ+μ} ↦ 1_{λ} ⊗ u + lower order terms ,

where U is a finite-dimensional 𝔤-module, and u ∈ U[μ]. In this paper we study singularities of Φ_{λ}^{u} as a function of λ and u.

The intertwining operator Φ_{λ}^{u} is completely determined by the image of the highest weight vector, which we denote by Sing(1_{λ} ⊗ u). For any u ∈ U[μ], the singular vector Sing(1_{λ} ⊗ u) can be written in terms of the Shapovalov form on M_{λ}, and is unique for generic λ. It is a rational function of λ with possible poles if λ belongs to at least one of the Kac-Kazhdan hyperplanes ⟨α, λ + ρ⟩ = \frac{k}{2}⟨α, α⟩. For those special values of λ the operator Φ_{λ}^{u} may fail to exist.

One of the objectives of this paper is to regularize the family of operators Φ_{λ}^{u}, that is, to construct a family of intertwining operators Φ̃_{λ}^{u} : M_{λ+μ} → M_{λ} ⊗ U, holomorphically depending on λ and u, such that the correspondence u ↦ Φ̃_{λ}^{u} is still injective.

We also study the relation between the operators Φ_{λ}^{u} for different values of λ and u. Recall that for a dominant integral weight ν and any element w of the Weyl group W, the Verma module M_{ν} contains a submodule isomorphic to M_{w·ν}. We fix this identification and regard M_{w·ν} as a submodule of M_{ν}.

If λ+μ is a dominant integral weight, then for any w ∈ W the restriction of the operator Φ_{λ}^{u} to the submodule M_{w·(λ+μ)} takes values in the submodule M_{w·λ} ⊗ U. The relation between this restriction and the operator Φ_{w·λ}^{u} is given by

1 Supported in part by RFFI grant 02-01-00085a and CRDF grant RM1-2334-MO-02.

2 Supported in part by NSF grant DMS-9801582.

Φ_{λ}^{u}|_{M_{w·λ}} = Φ_{w·λ}^{A_{w}(λ)u} ,

where A_{w}(λ) is a rational End(U)-valued function of λ. The operators A_{w}(λ), w ∈ W, are called the dynamical Weyl group operators.

The specialization of the dynamical Weyl group operators at λ = −ρ produces constant operators w̄ = A_{w}(−ρ), and for a simple root reflection s_{i} ∈ W we have s̄_{i} = f_{i}^{h_{i}}. According to the classical results of Verma and Lusztig, the operators s̄_{i} form a representation of the Weyl group. The dynamical Weyl group operators may be regarded as meromorphic deformations of the operators w̄ in the class of cocycles on the Weyl group, satisfying

A_{w_{1}w_{2}}(λ) = A_{w_{1}}(w_{2} · λ)A_{w_{2}}(λ) .

We introduce a notion of a regularizing operator N(λ), acting in U and polynomially depending on λ. By definition, we require that N(λ) has the following properties.

First, the modified intertwining operators Φ_{λ}^{N(λ)u} : M_{λ+μ} → M_{λ} ⊗ U are holomorphic in λ for any u ∈ U[μ].

Second, the conjugated dynamical Weyl group operators N(w · λ)^{−1}A_{w}(λ)N(λ) are constant operators. This implies, in particular, that the cocycle A_{w}(λ) is cohomologically equivalent to a constant function of λ.

Third, we impose a minimality condition on N(λ) by requiring that det N(λ) is equal to a certain explicitly specified polynomial. (The second property of N(λ) implies the divisibility of det N(λ) by this polynomial.)

We prove that the regularizing operator N(λ), if it exists, is uniquely determined up to the right multiplication by an operator, polynomially depending on λ, invertible for all values of λ, and symmetric with respect to the Weyl group action. If 𝔤 = 𝔰𝔩_{2} it is easy to check that the regularizing operator exists for any irreducible finite-dimensional 𝔤-module U; in each weight subspace of U the operator N(λ) acts as a scalar, determined by the determinant condition. In this paper we prove the existence property in the case 𝔤 = 𝔰𝔩_{3}. We conjecture that a regularizing operator exists for any simple Lie algebra 𝔤 and any finite-dimensional 𝔤-module U.

Our construction of the regularizing operator N(λ) relies on a certain realization of finitedimensional 𝔰𝔩_{3}-modules, based on a special case of the (𝔤𝔩_{m}, 𝔤𝔩_{n}) duality. We identify weight subspaces U[μ] of an 𝔰𝔩_{3}-module U with subspaces of singular vectors in tensor products V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)} of finite-dimensional 𝔤𝔩_{2}-modules. In this realization the action of operators A_{w}(λ) is identified with the action of suitably renormalized standard rational Rmatrices, the spectral parameters in the R-matrices being determined by λ.

We use the functional realization H[l] of the tensor product V_{(l_{1},0)}⊗V_{(l_{2},0)}⊗V_{(l_{3},0)}, provided by the representation theory of the Yangian Y (𝔤𝔩_{2}), to define operators N [l; z] : H[l] → V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)}, which conjugate the renormalized R-matrices to the identity. We then use results on the reducibility of the tensor product V_{(l_{1},0)}(z_{1}) ⊗ V_{(l_{2},0)}(z_{2}) ⊗ V_{(l_{3},0)}(z_{3}) of the evaluation Y (𝔤𝔩_{2})-modules, to prove that for any u ∈ Im N [l; z] the operator Φ_{λ}^{u} is holomorphic in λ.

The final step of the construction is an alternative ‘constant’ identification of the functional space H[l]^{sing} with the subspace of singular vectors (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing}, which is provided by a certain combinatorial lemma. The composition of the ‘constant’ identification with N [l; z] gives the desired regularizing operator N(λ) in our particular realization.

As an application of our results, we use the regularized family of intertwining operators to introduce an End(U[0])-valued function Ψ(λ, x) of λ ∈ 𝔥^{∗}, x ∈ 𝔥, defined as a certain matrix trace of Φ_{λ}^{u}. This Baker-Akhiezer type function is holomorphic in λ, and satisfies certain algebraic identites (resonance conditions), relating values of Ψ(λ, x) for different values of λ. The fact that N(λ) conjugates the dynamical Weyl group operators A_{w}(λ) to the constant operators w̄ implies that these identities have a simple form

Ψ(λ, x)u = Ψ(s_{α} · λ, x)u ,

for suitable λ, u. Special cases of such resonance conditions were used in [ES] to establish the algebraic integrability of the generalized quantum Calogero-Sutherland systems.

This paper is organized as follows. In Section 2 we review facts on 𝔤-modules and set the notation. In Section 3 we recall the construction of intertwining operators Φ : M_{λ+μ} → M_{λ} ⊗ U and singular vectors in the tensor product.

In Section 4 we review facts about the Weyl group and the dynamical Weyl group. We introduce a distinguished action of W in U, and show that the dynamical Weyl group operators can be viewed as its deformation. We establish Theorems 4 and 5, which improve results of [EV, TV].

In Section 5 we introduce the notion of a regularizing operator N(λ), and use the axiomatic description to establish properties of regularizing operators. Theorem 9 gives the uniquiness of a regularizing operator. We formulate Theorem 10 on the existence of regularizing operators for 𝔤 = 𝔰𝔩_{3}. This is the main technical result of the paper.

Section 6 provides the background necessary to construct regularizing operators for 𝔰𝔩_{3}. We invoke the (𝔤𝔩_{m}, 𝔤𝔩_{n})-duality and functional spaces to give a realization of an 𝔰𝔩_{3}-module U, and use the realization to construct the fundamental operator N [l; z], which is the main ingredient in the construction of N(λ).

Section 7 contains the construction of regularizing operators for 𝔰𝔩_{3} and proofs of all their properties. The short Section 8 is devoted to resonance conditions for trace functions, renormalized by N(λ).

In Appendix A we review the calculus of formal monomials, which is used to prove the properties of the dynamical Weyl group operators. Appendix B is devoted to the representation theory of the Yangian Y (𝔤𝔩_{2}). We use Yangian modules to establish properties of the operator N [l; z]. Appendix C contains a proof of a technical combinatorial Lemma 19, which is used in the main construction.

2. Notation

Let 𝔤 be a simple Lie algebra over ℂ with a Cartan subalgebra 𝔥 and a root system Δ with a polarization Δ = Δ^{+} ∪ Δ^{−}. We identify 𝔥 with 𝔥^{∗} using the Killing form on 𝔤, and denote the induced invariant bilinear form on 𝔥^{∗} by ⟨·, ·⟩.

An element λ ∈ 𝔥^{∗} is called dominant, if ⟨α, λ⟩ ≥ 0 for any α ∈ Δ^{+}. We write λ ≥ μ for λ, μ ∈ 𝔥^{∗}, if λ − μ is dominant. This is a partial order on 𝔥^{∗}.

An element λ ∈ 𝔥^{∗} is called integral, if \frac{2⟨α,λ⟩}{⟨α,α⟩} ∈ ℤ for any α ∈ Δ^{+}.

Denote W the Weyl group of 𝔤. We introduce two actions of W in 𝔥^{∗}. The standard action is defined by

s_{α}v = v − \frac{2⟨α, v⟩}{⟨α, α⟩}α

for any root reflection s_{α} ∈ W, and the “dot” action is defined by

s_{α} · v = s_{α}(v + ρ) − ρ,

where ρ = \frac{1}{2} ∑_{α∈Δ^{+}} α.

Let {e_{i}, h_{i}, f_{i}} denote the standard generators of the Lie algebra 𝔤, corresponding to a simple root α_{i}. We denote 𝔰𝔩_{2}(α_{i}) the Lie subalgebra of 𝔤, spanned by e_{i}, h_{i}, f_{i}.

We denote 𝔫^{+}, 𝔫^{−} the Lie subalgebras, generated respectively by {e_{i}} and {f_{i}}. We have a vector space decomposition 𝔤 = 𝔫^{+} ⊕ 𝔥 ⊕ 𝔫^{−}.

The root lattice Q is defined by Q = ∑_{i=1}^{dim 𝔥} ℤα_{i}. We also set Q^{+} = ∑_{i=1}^{dim 𝔥} ℤ_{≥0}α_{i}.

The universal enveloping algebra U(𝔤) is a Q-graded associative algebra, with the grading defined by

wt(e_{i}) = α_{i}, wt(h_{i}) = 0, wt(f_{i}) = −α_{i}, i = 1, …, dim 𝔥.

The Poincare-Birkhoff-Witt theorem gives us a decomposition

U(𝔤) = U(𝔫^{−}) ⊗ U(𝔥) ⊗ U(𝔫^{+}),

and we denote π_{0} : U(𝔤) → U(𝔥) the induced projection along the subspace (𝔫^{−}U(𝔤) + U(𝔤)𝔫^{+}) ⊂ U(𝔤).

Introduce an anti-involution ϖ of 𝔤 by

ϖ(e_{i}) = f_{i},  ϖ(f_{i}) = e_{i},  ϖ(h_{i}) = h_{i},  i = 1, …, dim 𝔥.

Define a bilinear U(𝔥)-valued form S on U(𝔤), by

S(x, y) = π_{0}(ϖ(x)y) ∈ U(𝔥), x, y, ∈ U(𝔤).

The form S is contravariant, i.e. satisfies

S(xy, z) = S(y, ϖ(x)z), x, y, z ∈ U(𝔤).

We will identify the commutative algebra U(𝔥) with the algebra ℂ[𝔥^{∗}] of polynomial functions on 𝔥^{∗}, and for any λ ∈ 𝔥^{∗} we will denote S_{λ} the ℂ-valued contravariant form on U(𝔤), obtained by evaluating S at λ.

We consider 𝔤-modules V with a weight space decomposition

V = ⨁_{μ∈𝔥^{∗}} V [μ], V [μ] = {v ∈ V | hv = μ(h)v for all h ∈ 𝔥},

with finite-dimensional weight subspaces V [μ]. We say that v ∈ V is homogeneous of weight μ and write wt(v) = μ, if v ∈ V [μ]. Then for any homogeneous x ∈ U(𝔤), we have wt(xv) = wt(x) + wt(v).

In this paper U always denotes a finite-dimensional 𝔤-module.

For any λ ∈ 𝔥^{∗}, we denote M_{λ} the Verma module for 𝔤, generated by the highest weight vector 1_{λ}, with relations

𝔫^{+}1_{λ} = 0, h1_{λ} = λ(h)1_{λ}, for h ∈ 𝔥.

We identify any Verma module M_{λ} with U(𝔫^{−}) by sending x ∈ U(𝔫^{−}) to x1_{λ} ∈ M_{λ}. The Verma module M_{λ} carries a bilinear 𝔤-contravariant form (·|·), defined by

(x1_{λ}|y1_{λ}) = S_{λ}(x, y), x, y ∈ U(𝔤).

This form is called the Shapovalov form. Different weight subspaces of M_{λ} are orthogonal with respect to this form, and its restriction to a weight subspace M_{λ}[μ] is nondegenerate unless λ belongs to a hyperplane of the form

χ_{α,k}(λ) =^{def} \frac{2⟨α, λ + ρ⟩}{⟨α, α⟩} − k = 0

for some α ∈ Δ^{+}, k ∈ ℤ_{≥0}.

The Verma module M_{λ} is reducible if and only if λ belongs to at least one of those hyperplanes. The kernel of the Shapovalov form is the maximal proper submodule of M_{λ}. Let V_{λ} denotes the irreducible quotient of the Verma module M_{λ}; it inherits a contravariant form and a weight decomposition from M_{λ}; moreover, the contraviriant form on V_{λ} is nondegenerate. It is known that the module V_{λ} is finite-dimensional if and only if λ is a dominant integral weight.

3. Singular vectors and intertwining operators

A vector v in a 𝔤-module V is called singular, if 𝔫^{+}v = 0. The subspace of all singular vectors in V is denoted V^{sing}. For any λ ∈ 𝔥^{∗}, we have an isomorphism

Hom_{𝔤}(M_{λ}, V) ≅ V^{sing}[λ],

constructed by associating with any intertwining operator Φ ∈ Hom_{𝔤}(M_{λ}, V) the image of the vector 1_{λ}.

It is known [BGG] that a Verma module M_{λ} contains a unique up to proportionality singular vector of weight ν if and only if there exists a finite sequence of weights

λ^{(0)} = λ, λ^{(1)}, …, λ^{(k)} = ν,

such that

λ^{(i)} = s_{β_{i}} · λ^{(i−1)} = λ^{(i−1)} − n_{i}β_{i}, i = 1, …, k,

for some β_{i} ∈ Δ^{+}, n_{i} ∈ ℤ_{≥0}. We write ν ≺ λ if this condition is satisfied.

Theorem 1. [FFM] Let λ ∈ 𝔥^{∗}, and let w ∈ W be such that w · λ ≺ λ. Let w = s_{i_{1}} … s_{i_{l}} be the reduced decomposition of w in terms of simple root reflections. Then the formal monomial

F_{w}(λ) = f_{i_{1}}^{γ_{1}(λ)} … f_{i_{l}}^{γ_{l}(λ)}, γ_{k}(λ) = ⟨α_{i_{k}}, (s_{k+1} … s_{l}) · λ⟩,

makes sense, and F_{w}(λ)1_{λ} is a singular vector of weight w · λ in the Verma module M_{λ}.

If λ is a dominant integral weight, then w · λ ≺ λ for any w ∈ W, and γ_{k}(λ) ∈ ℤ_{≥0} for all k = 1, …, l. Therefore, F_{w}(λ) is a well-defined element in U(𝔫^{−}). The meaning of F_{w}(λ) for other λ ∈ 𝔥^{∗}, when we may have γ_{k}(λ) ∉ ℤ_{≥_{0}}, is explained in Appendix A.

Introduce notation v_{w·λ} = F_{w}(λ)1_{λ} for the singular vector, constructed in Theorem 1. We identify the submodule of M_{λ}, generated by v_{w·λ}, with M_{w·λ} by mapping 1_{w·λ} to v_{w·λ}.

We now study singular vectors in tensor products M_{λ} ⊗ U, λ ∈ 𝔥^{∗}. All tensor products in this paper are considered over the field ℂ.

Let {g_{i}} be a homogeneous basis in U(𝔫^{−}), and let (S_{λ}^{−1})_{ij} be the inverse to the matrix of the form S_{λ}, restricted to U(𝔫^{−}), with respect to this basis. The matrix elements (S_{λ}^{−1})_{ij} are rational functions of λ ∈ 𝔥^{∗}, with possible simple poles at the hyperplanes χ_{α,k}(λ) = 0, for some α ∈ Δ^{+} and k ∈ ℤ_{≥0}.

Introduce the Cartan anti-automorphism ω of the Lie algebra 𝔤 by

ω(e_{i}) = −f_{i},  ω(f_{i}) = −e_{i},  ω(h_{i}) = h_{i},  i = 1, …, dim 𝔥 .

Consider a rational U(𝔫^{−}) ⊗ End_{ℂ}(U)-valued function of λ ∈ 𝔥^{∗},

Ξ(λ) = ∑_{i,j} (S_{λ}^{−1})_{ij} g_{i} ⊗ ω(g_{j}) ,  (3.1)

where we regard ω(g_{j}) as operators acting in U. Note that the summation over i, j is finite, because (S_{λ}^{−1})_{ij} = 0 for sufficiently large i and ω(g_{j}) acts as zero for sufficiently large j. One can check that Ξ(λ) does not depend on the choice of the basis {g_{j}}.

For any λ ∈ 𝔥^{∗} and u ∈ U such that Ξ(λ)(1 ⊗ u) ∈ U(𝔫^{−}) ⊗ U is well-defined, we denote

Sing(1_{λ} ⊗ u) = Ξ(λ)(1_{λ} ⊗ u) ∈ M_{λ} ⊗ U.

For generic λ, the vector Sing(1_{λ} ⊗ u) is the unique singular vector of the form

Sing(1_{λ} ⊗ u) = 1_{λ} ⊗ u + lower order terms ,

where the “lower order terms” is a linear combination of vectors v′ ⊗ u′ with wt(v′) < λ, see [ES]. The vector Sing(1_{λ} ⊗ u) is a singular vector for any λ, when it is well-defined.

For any u ∈ U[μ] such that Sing(1_{λ} ⊗ u) is well defined we introduce an intertwining operator Φ_{λ}^{u} ∈ Hom_{𝔤}(M_{λ+μ}, M_{λ} ⊗ U), uniquely determined by the condition

Φ_{λ}^{u}1_{λ+μ} = Sing(1_{λ} ⊗ u).

4. Weyl group and dynamical Weyl group

The Weyl group W, associated with 𝔤, is generated by simple reflections {s_{i}}, i = 1, …, dim 𝔥, subject to the relations s_{i}^{2} = id, and the braid relations

\underbrace{s_{i}s_{j}s_{i} …}_{n_{ij} factors} = \underbrace{s_{j}s_{i}s_{j} …}_{n_{ij} factors},  (4.1)

for i ≠ j, where n_{ij} = 2, 3, 4, 6 if \frac{4⟨α_{i},α_{j}⟩^{2}}{⟨α_{i},α_{i}⟩⟨α_{j},α_{j}⟩} = 0, 1, 2, 3 respectively.

The length l(w) of any w ∈ W is defined to be the smallest positive integer l such that w can be written as w = s_{i_{1}} … s_{i_{l}} for some i_{1}, …, i_{l}. For the identity element id ∈ W we set l(id) = 0.

Let U be a finite-dimensional 𝔤-module. For any i = 1, …, dim 𝔥, introduce an operator s̄_{i} ∈ End_{ℂ}(U) by setting

s̄_{i}f_{i}^{k}v = f_{i}^{n−k}v,  k = 0, 1, …, n,  (4.2)

for any v ∈ U such that e_{i}v = 0 and h_{i}v = nv for some n ∈ ℤ_{≥0}. One can think of operators s̄_{i} as operators f_{i}^{h_{i}}.

The following lemma is well-known; see [EV] and references therein.

Lemma 2. There exists a linear representation w ↦ w̄ of the Weyl group W in U, such that the generators s_{i} are mapped to the operators s̄_{i}.

Obviously, for any w ∈ W we have

w̄ = ⨁ w̄_{μ},  w̄_{μ} ∈ Hom_{ℂ}(U[μ], U[wμ]).

Note that if w ∈ W and μ ∈ 𝔥^{∗} are such that wμ = μ, then w̄_{μ} = Id_{U[μ]}. One can show that the equivalence class of the representation w ↦ w̄ of W is completely determined by these conditions.

Remark. The operators w̄ do not preserve the Shapovalov form on U; one might try to modify the formula (4.2), and introduce operators s̃_{i} by

s̃_{i}f_{i}^{k}v = \frac{k!}{(n − k)!}f_{i}^{n−k}v,  k = 0, 1, …, n,  (4.3)

for any v ∈ U such that e_{i}v = 0 and h_{i}v = nv for some n ∈ ℤ_{≥0}. The new operators s̃_{i} clearly preserve the Shapovalov form.

Conjecture 3. There exists a representation of the Weyl group W in U, such that the generators s_{i} are mapped to the operators s̃_{i}.

This conjecture can be verified in certain cases, including adjoint representations for arbitrary 𝔤; however, no general proof is available.

We now study the dynamical Weyl group, introduced in [TV], [EV]. The following Theorem 4 and Theorem 5 are refinements of previous results.

Let ℂ[𝔥^{∗}], ℂ(𝔥^{∗}) denote respectively the algebras of polynomial and rational functions on 𝔥^{∗}. If V is a vector space and F ∈ V ⊗ ℂ[𝔥^{∗}], we denote F (λ) ∈ V the value of the V -valued polynomial F at point λ ∈ 𝔥^{∗}.

Theorem 4. For any w ∈ W there exists an operator A_{w} ∈ End_{ℂ}(U) ⊗ ℂ(𝔥^{∗}), satisfying the following property.

Suppose λ, μ ∈ 𝔥^{∗} are such that w · (λ + μ) ≺ λ + μ, and u ∈ U[μ] is such that Sing(1_{λ} ⊗ u) is well-defined. Then A_{w}(λ)u is well-defined, and

1. If w · λ ≺ λ, then

Φ_{λ}^{u}v_{w·(λ+μ)} = v_{w·λ} ⊗ A_{w}(λ)u + lower order terms .  (4.4)

2. If the condition w · λ ≺ λ fails, then A_{w}(λ)u = 0.

This theorem is proved in Appendix A.

The collection of operators A_{w} ∈ End_{ℂ}(U) ⊗ ℂ(𝔥^{∗}), w ∈ W, is called the dynamical Weyl group. Equation (4.4) implies that

A_{w} = ⨁_{μ} A_{w}^{μ}, A_{w}^{μ} ∈ Hom_{ℂ}(U[μ], U[wμ]) ⊗ ℂ(𝔥^{∗}).

Example. Let U be a finite-dimensional 𝔰𝔩_{2}-module, generated by a highest weight vector v of weight n ∈ ℤ_{≥0}. Then the dynamical Weyl group consists of two operators: A_{id} ≡ Id and A_{s_{1}}, determined by

A_{s_{1}}(λ)f^{k}v = (−1)^{n} \frac{(−t − 2)(−t − 3) … (−t − n + k − 1)}{t(t − 1) … (t − k + 1)} \frac{k!}{(n − k)!}f^{n−k}v, k = 0, 1, …, n , (4.5)

where t = (λ, α_{1}).

For arbitrary 𝔤, the operators A_{s_{i}}(λ) act in any irreducible 𝔰𝔩_{2}(α_{i}) submodule of U by formula (4.5), with t = \frac{2⟨λ,α_{i}⟩}{⟨α_{i},α_{i}⟩}.

Theorem 5. The dynamical Weyl group operators A_{w}, w ∈ W, acting in a finite-dimensional 𝔤-module U, have the following properties.

1. The dynamical Weyl group operators satisfy the cocycle condition,

A_{w_{1}w_{2}}(λ) = A_{w_{1}}(w_{2} · λ)A_{w_{2}}(λ),  w_{1}, w_{2} ∈ W.  (4.6)

2. The operator A_{w}, regarded as an End_{ℂ}(U)-valued rational function on 𝔥^{∗}, has at most simple poles, which may occur only at the hyperplanes

χ_{α,k}(λ) = 0, α ∈ Δ^{+} ∩ w(Δ^{−}), k ∈ ℤ_{≥0}.

3. If we identify U[μ] and U[wμ] in any way not depending on λ, then for generic λ ∈ 𝔥^{∗} we have

det A_{w}^{μ}(λ) = const ∏_{α∈Δ^{+}∩w(Δ^{−})} \frac{∏_{k=0}^{∞} χ_{α,k}(w · λ)^{dim U[wμ+kα]}}{∏_{k=0}^{∞} χ_{α,k}(λ)^{dim U[μ+kα]}} .

where the constant depends on the choice of identification.

Remark. The dynamical Weyl group operators A_{w}(λ) were introduced in [TV], [EV], using a different normalization of the singular vectors:

v_{w·λ} = \frac{f_{i_{1}}^{n_{1}}}{n_{1}!} … \frac{f_{i_{l}}^{n_{l}}}{n_{l}!} 1_{λ}, n_{k} = ⟨α_{i_{k}}, (s_{k+1} … s_{l}) · λ⟩.

Equation (4.4) was observed in [TV], [EV] only for λ ≫ μ. Part 2 of Theorem 4 is new.

In [TV], [EV], the cocycle condition was satisfied only for w_{1}, w_{2} ∈ W such that l(w_{1}w_{2}) = l(w_{1}) + l(w_{2}); equivalently, operators A_{w}(λ) formed a cocycle on the braid group. Our operators satisfy the additional conditions A_{s_{α}}(s_{α} · λ)A_{s_{α}}(λ) = Id, and thus form a cocycle on the Weyl group.

Proof of Theorem 5. It follows directly from the construction (see also [TV, EV]) that the cocycle condition (4.6) is satisfied when w_{1}, w_{2} ∈ W are such that l(w_{1}w_{2}) = l(w_{1}) + l(w_{2}). Therefore, it suffices to show that

A_{s_{i}}(s_{i} · λ)A_{s_{i}}(λ) = Id, i = 1, …, dim 𝔥.

This follows from the explicit formula (4.5) for the action of A_{s_{i}}(λ) in any irreducible 𝔰𝔩_{2}(α_{i})- submodule of U.

Now, let w ∈ W have a reduced decomposition w = s_{i_{1}} … s_{i_{l}}. Then the cocycle condition implies

A_{w}(λ) = A_{s_{i_{1}}}(s_{i_{2}} … s_{i_{l}} · λ) … A_{s_{i_{l}}}(λ),

and singularities of A_{w}(λ) are determined by the singularities of A_{s_{i_{k}}} (s_{i_{k+1}} … s_{i_{l}} · λ) for k = 1, …, dim 𝔥. From the explicit formula (4.5) it follows that A_{s_{i_{k}}} (s_{i_{k+1}} … s_{i_{l}} · λ) may only have simple poles at the hyperplanes χ_{α_{i_{k}} ,n}(s_{i_{k+1}} … s_{i_{l}} · λ) = 0, or equivalently

χ_{β_{k},n}(λ) = 0,  β_{k} = s_{i_{l}} … s_{i_{k+1}}α_{i_{k}}.  (4.7)

It is known that if w = s_{i_{1}} … s_{i_{l}} is a reduced expression, then {β_{k}} form (without repetitions) the set Δ^{+} ∩ w(Δ^{−}). Therefore, the hyperplanes (4.7) are all distinct, and the second assertion follows.

Finally, from the cocycle property we obtain

det A_{w}^{μ}(λ) = det A_{s_{i_{1}}}(s_{i_{2}} … s_{i_{l}} · λ) … det A_{s_{i_{l}}}(λ)

and the proof of the general formula reduces to verification of the formula

det A_{s_{i}}^{μ}(λ) = const \frac{∏_{k=0}^{∞} χ_{α_{i},k}(s_{i} · λ)^{dim U[s_{i}μ+kα_{i}]}}{∏_{k=0}^{∞} χ_{α_{i},k}(λ)^{dim U[μ+kα_{i}]}} .

which follows from (4.5).

The cocycle condition (4.6) immediately implies that the operators A_{w}(−ρ), w ∈ W, form a representation of the Weyl group. One can easily check that in fact A_{w}(−ρ) = w̄, where operators w̄ are defined by (4.2).

5. Regularizing operators and their properties

In this section we define the main object of our study — regularizing operators for finitedimensional representations of a semisimple Lie algebra 𝔤, and establish some of their properties.

Let 𝔤 be a simple Lie algebra, and let U be a finite-dimensional 𝔤-module. An operator N ∈ End_{ℂ}(U) ⊗ ℂ[𝔥^{∗}] is called a regularizing operator, if it satisfies the following conditions.

1. The operator N preserves weight subspaces, i.e. it can be decomposed as

N = ⨁_{μ} N_{μ}, N_{μ} ∈ End_{ℂ}(U[μ]) ⊗ ℂ[𝔥^{∗}].

2. For any μ ∈ 𝔥^{∗}, there exists a nonzero constant c_{μ} such that

det N_{μ}(λ) = c_{μ} ∏_{α∈Δ^{+}} ∏_{k=0}^{∞} χ_{α,k}(λ)^{dim U[μ+kα]}.  (5.1)

3. There exist operators a_{w} ∈ End_{ℂ}(U), such that for any w ∈ W and generic λ ∈ 𝔥^{∗} we have

a_{w} = N(w · λ)^{−1}A_{w}(λ)N(λ).  (5.2)

4. The operator Ξ_{N}(λ) : U → U(𝔫^{−}) ⊗ U, defined by

Ξ_{N}(λ)u = Ξ(λ)(1 ⊗ N(λ)u), u ∈ U,

depends polynomially on λ. Here Ξ(λ) is given by (3.1). Remark. The last condition is equivalent to the requirement that for any u ∈ U, the

singular vector Sing(1_{λ} ⊗ N(λ)u) ∈ M_{λ} ⊗ U is polynomial in λ.

The following Lemma is convenient for establishing the formula (5.1).

Lemma 6. Suppose N ∈ End_{ℂ}(U) ⊗ ℂ[𝔥^{∗}] satisfies conditions (1) and (3) above, and is such that N^{−1} is regular outside the hyperplanes χ_{α,k}(λ) = 0 for all α ∈ Δ^{+} and k ∈ ℤ_{≥0}. Then N also satisfies condition (2).

Proof. The formula (5.2) implies that for any w ∈ W and generic λ ∈ 𝔥^{∗}, we have

det A_{w}^{μ}(λ) det N_{μ}(λ) = const det N_{wμ}(w · λ).  (5.3)

The right-hand side of this equation polynomially depends on λ, and therefore in the lefthand side det N_{μ}(λ) must be divisible by the denominator of det A_{w}^{μ}(λ). In the special case when w = w_{0} is the longest element of the Weyl group, this denominator is precisely ∏_{α∈Δ^{+}} ∏_{k=0}^{∞} χ_{α,k}(λ)^{dim U[μ+kα]}. Therefore, we conclude that

c_{μ}(λ) = \frac{det N_{μ}(λ)}{∏_{α∈Δ^{+}} ∏_{k=0}^{∞} χ_{α,k}(λ)^{dim U[μ+kα]}}

is polynomial in λ, and (5.3) implies that c_{μ}(λ) = const c_{w_{0}μ}(w_{0} · λ).

By the assumption of the Lemma, c_{μ}(λ) may only vanish at hyperplanes χ_{α,k}(λ) = 0, and similarly c_{w_{μ}}(w_{0} · λ) may only vanish at hyperplanes χ_{α,k}(w_{0} · λ) = 0. Since these two families of hyperplanes are disjoint, we conclude that c_{μ}(λ) is a polynomial in λ which never vanishes, i.e. a constant polynomial. The condition (2) follows.

The argument above also shows that the condition (2) can be thought of as a minimality condition for N.

Here are some other immediate corollaries of the definition.

Proposition 7. Let U be a finite-dimensional 𝔤-module, and let N be a regularizing operator.

1. The operators a_{w} ∈ End_{ℂ}(U) form a representation of the Weyl group.

2. The inverse operator N^{−1}, regarded as an End_{ℂ}(U)-valued rational function on 𝔥^{∗}, may have only simple poles. The poles may only occur at hyperplanes χ_{α,k}(λ) = 0 for α ∈ Δ^{+} and k ∈ ℤ_{≥0} such that U[μ + kα] ≠ 0.

3. Let C ∈ End_{ℂ}(U) be a weight-preserving operator, i.e.

C = ⨁_{μ} C_{μ}, C_{μ} ∈ End(U[μ]).

Then the operator N̄ = NC is also a regularizing operator.

Proof. For any w_{1}, w_{2} ∈ W, we compute

a_{w_{1}w_{2}} = N(w_{1}w_{2} · λ)^{−1}A_{w_{1}w_{2}}(λ)N(λ) = N(w_{1}w_{2} · λ)^{−1}A_{w_{1}}(w_{2} · λ)A_{w_{2}}(λ)N(λ) =
(N(w_{1}w_{2} · λ)^{−1}A_{w_{1}}(w_{2} · λ)N(w_{2} · λ)) (N(w_{2} · λ)^{−1}A_{w_{2}}(λ)N(λ)) = a_{w_{1}}a_{w_{2}},

which shows that the correspondence w ↦ a_{w} gives a representation of the Weyl group.

From the formula for det N(λ), we see that N^{−1} may only have poles at hyperplanes χ_{α,k}(λ) = 0, for some α ∈ Δ^{+} and k ∈ ℤ_{≥0} such that U[μ + kα] ≠ 0. Fix α ∈ Δ_{+} and k ∈ ℤ_{≥0}. We have

N(λ)^{−1} = a_{s_{α}}^{−1}N(s_{α} · λ)^{−1}A_{s_{α}}(λ) .

since N(s_{α} · λ)^{−1} does not have a pole at the hyperplane χ_{α,k}(λ) = 0, the singularity of N(λ)^{−1} may come only from the operator A_{s_{α}}(λ), which have at most simple pole there.

The operators N̄(λ) = N(λ)C are obviously polynomial in λ, and the conditions on C_{μ} ensure weight preserving and determinant properties of N̄(λ). Finally,

ā_{w} = N̄(w · λ)^{−1}A_{w}(λ)N̄(λ) = C^{−1}N(w · λ)^{−1}A_{w}(λ)N(λ)C = C^{−1}a_{w}C

do not depend on λ, and for any u ∈ U the vector

ΞN̄(λ)u = Ξ(λ)(1 ⊗ N̄(λ)u) = Ξ(1 ⊗ N(λ)Cu) = Ξ_{N}(λ)(Cu)

is polynomial in λ.

Theorem 8. Let U be a finite-dimensional 𝔤-module, and let N ∈ End_{ℂ}(U) ⊗ ℂ[𝔥^{∗}] be a regularizing operator. Fix any λ_{0} ∈ 𝔥^{∗}. Then 1. The linear map

Ξ_{N}(λ_{0}) : U → (M_{λ_{0}} ⊗ U)^{sing} , u ↦ Sing(1_{λ_{0}} ⊗ N(λ_{0})u)

is injective.

2. We have the following description of the image of N_{μ}(λ_{0}):

Im N(λ_{0}) = { u ∈ U | Sing(1_{λ_{0}} ⊗ u) is well-defined } .

Proof. Let Δ_{0} be the root subsystem of Δ, defined by

Δ_{0} = { α ∈ Δ | ⟨α, λ_{0}⟩ ∈ ℤ} .

We set Δ_{0}^{+} = Δ_{0} ∩ Δ^{+}, and denote by W_{0} the subgroup of W, generated by root reflections s_{α}, α ∈ Δ_{0}.

Let w ∈ W_{0} be such that w·λ_{0} is antidominant with respect to Δ_{0}^{+}. Then, det N(w·λ_{0}) ≠ 0, and N(w · λ_{0}) is invertible.

According to Theorem 4,

Φ_{λ_{0}}^{N(λ_{0})u}v_{w·(λ_{0}+μ)} = v_{w·λ_{0}} ⊗ A_{w}(λ_{0})N(λ_{0})u + lower order terms .

Suppose for some u ∈ U[μ] we have Sing(1_{λ_{0}} ⊗ N(λ_{0}))u = 0. Then the intertwining operator Φ_{λ_{0}}^{N(λ_{0})u} is identically zero, and in particular

N(w · λ_{0})a_{w}u = A_{w}(λ_{0})N(λ_{0})u = 0.

Since N(w · λ_{0}) and a_{w} are both invertible, we conclude that u = 0, and thus Ξ_{N}(λ_{0}) is injective.

Now, let u ∈ U[μ] be such that Sing(1_{λ_{0}} ⊗u) is well-defined. Then the vector u′ = A_{w}(λ_{0})u is also well-defined, and we have

u = A_{w}(λ_{0})^{−1}u′ = N(λ_{0})a_{w}^{−1}N(w · λ_{0})^{−1}u′ ∈ Im N(λ_{0}).

Conversely, property (4) of N(λ_{0}) guarantees that Sing(1_{λ_{0}} ⊗ u) is well-defined for any u ∈ Im N_{μ}(λ_{0}).

Finally, we prove that the regularizing operator is determined uniquely up to right multiplication by a polynomially invertible matrix C(λ), satisfying certain W -invariance conditions.

Theorem 9. Let N, N̄ be regularizing operators, and let a_{w}, ā_{w} be the associated constant representations of the Weyl group. Then the operator

C(λ) = N(λ)^{−1}N̄(λ) ∈ End_{ℂ}(U)

is polynomial in λ, has a decomposition

C(λ) = ⨁_{μ} C_{μ}(λ), C_{μ}(λ) ∈ End_{ℂ}(U[μ]),

and satisfies

C(w · λ)ā_{w} = a_{w}C(λ).

Moreover, the inverse operator C(λ)^{−1} is also polynomial in λ, and the Weyl group representations a_{w}, ā_{w} are equivalent.

Proof. Since regularizing operators preserve weight subspaces, we have

C_{μ}(λ) = N_{μ}(λ)^{−1}N̄_{μ}(λ) ∈ End_{ℂ}(U[μ]).

The only possible singularities of C(λ) = N(λ)^{−1}N̄(λ) may be simple poles at one of the hyperplanes χ_{α,k}(λ) = 0. Fix α ∈ Δ^{+}, k ∈ ℤ_{≥0}, and write

N(λ)^{−1} = \frac{X(λ)}{χ_{α,k}(λ)} + Y (λ)

for some X(λ), Y (λ) ∈ End_{ℂ}(U), regular at the hyperplane χ_{α,k}(λ) = 0.

It follows from the identity N(λ)^{−1}N(λ) = Id that for generic λ_{0} from this hyperplane, we have X(λ_{0})N(λ_{0}) = 0. Hence,

ker X(λ_{0}) ⊃ Im N(λ_{0}) = Im N̄(λ_{0}) ,

which implies that C(λ) is well-defined at λ_{0}. Therefore, C(λ) does not have a pole at the hyperplane χ_{α,k}(λ) = 0, which was arbitrary, and thus C(λ) is polynomial in λ.

In view of (5.1), we conclude that det C_{μ}(λ) = (det N_{μ}(λ))^{−1} det N̄_{μ}(λ) is a nonzero constant. Hence, each C_{μ}(λ) is polynomially invertible.

Finally, we compute

C(w · λ)^{−1}a_{w}C(λ) = N̄(w · λ)^{−1}N(w · λ)a_{w}N(λ)^{−1}N̄(λ) = N̄(w · λ)^{−1}A_{w}(λ)N̄(λ) = ā_{w}.

The invariance condition follows. Setting λ = −ρ, we get

ā_{w} = C(−ρ)^{−1}a_{w}C(−ρ)

and the representations a_{w}, ā_{w} of the Weyl group are equivalent.

Example: Let 𝔤 = 𝔰𝔩_{2}. We identify 𝔥^{∗} with ℂ by associating 𝔥^{∗} ∋ λ ↔ ⟨α_{1}, λ⟩ ∈ ℂ. Under this identification, α_{1} ≡ 2 and ρ ≡ 1.

Let U be an irreducible finite-dimensional 𝔰𝔩_{2}-module with highest weight Λ ∈ ℤ_{≥0}. For u ∈ U[Λ − 2k], k ∈ ℤ_{≥0}, we have

Sing(1_{λ} ⊗ u) = ∑_{j=0}^{k} \frac{(−1)^{j}}{j! ∏_{j=0}^{k}(λ − j)}f^{j}1_{λ} ⊗ e^{j}u .

The weight subspaces of U are one-dimensional, the operators N_{Λ−2k}(λ) ∈ End(U[Λ − 2k]) ≅ ℂ being scalars, and we take

N_{Λ−2k}(λ) = \frac{1}{k!} ∏_{j=0}^{k−1} χ_{α,j}(λ) = \frac{1}{k!} ∏_{j=0}^{k−1}(λ − j) .

One can check that the operator a_{s} acts in U by

a_{s} : f^{k}1_{Λ} ↦ f^{Λ−k}1_{Λ}, k = 0, 1, …, Λ .

In particular, if Λ is an even integer and u = f^{Λ/2}1_{Λ} ∈ U[0], we have a_{s}u = u.

We now state our main result on the existence of regularizing operators.

Theorem 10. Let 𝔤 = 𝔰𝔩_{3}, and let U be an irreducible finite-dimensional 𝔤-module. Then there exists a regularizing operator N ∈ End_{ℂ}(U)⊗ℂ(𝔥^{∗}). Moreover, the associated operators a_{s_{i}}, corresponding to simple root reflections in W, coincide with the operators s̄_{i} given by the formulas (4.2).

This theorem will be proved in Section 7.

Example. Let U be the adjoint representation of 𝔰𝔩_{3}. The zero weight subspace U[0] is the Cartan subalgebra 𝔥; the generators h_{1}, h_{2} ∈ 𝔰𝔩_{3} form a basis in U[0].

The action of operators A_{w}^{[0]}(λ) on U[0] is completely determined by the matrices

A_{s_{1}}^{[0]}(λ) ( 0  1^{\frac{−λ_{1}−2}{λ_{1}}  \frac{−λ_{1}−1}{λ_{1}}}) ,  A_{s_{2}}^{[0]}(λ) ( 1  0_{\frac{−λ_{2}−1}{λ_{2}}  \frac{−λ_{2}−2}{λ_{2}}}) ,

where λ_{1} = ⟨α_{1}, λ⟩, λ_{2} = ⟨α_{2}, λ⟩. The operator N_{[0]}(λ) : U[0] → U[0] is given by the matrix

N_{[0]}(λ) = \frac{1}{3} (\begin{matrix}2λ_{1}^{2} + 2λ_{1}λ_{2} − λ_{2}^{2} + 4λ_{1} − λ_{2} & 2λ_{1}^{2} + 2λ_{1}λ_{2} − λ_{2}^{2} + 2λ_{1} − 2λ_{2} \\ 2λ_{2}^{2} + 2λ_{1}λ_{2} − λ_{1}^{2} − 2λ_{1} + 2λ_{2} & 2λ_{2}^{2} + 2λ_{1}λ_{2} − λ_{1}^{2} − λ_{1} + 4λ_{2} \end{matrix}) .

It is a straightforward computation to show that

N_{[0]}(s_{1} · λ)^{−1}A_{s_{1}}^{[0]}(λ)N_{[0]}(λ) ≡ Id_{U[0]}, N_{[0]}(s_{2} · λ)^{−1}A_{s_{2}}^{[0]}(λ)N_{[0]}(λ) ≡ Id_{U[0]},

and that

det N_{[0]}(λ) = λ_{1}λ_{2}(λ_{1} + λ_{2} + 1) = χ_{α_{1},1}(λ)χ_{α_{2},1}(λ)χ_{α_{1}+α_{2},1}(λ).

6. The (𝔤𝔩_{m}, 𝔤𝔩_{n}) duality, and functional realizations of 𝔤𝔩_{n}-modules

In Section 7 we will construct the regularizing operators for 𝔤 = 𝔰𝔩_{3} using a concrete realization of irreducible finite-dimensional 𝔰𝔩_{3}-modules. This section is devoted to a review of some underlying results on representations of the reductive Lie algebras 𝔤𝔩_{n}.

The space ℂ^{mn} = (ℂ^{m})^{⊗n} = (ℂ^{n})^{⊗m} has natural commuting actions of 𝔤𝔩_{m} and 𝔤𝔩_{n}. The algebra ℙ_{m,n} = S(ℂ^{mn}) of polynomial functions on ℂ^{mn} becomes a module over 𝔤𝔩_{m} ⊗ 𝔤𝔩_{n}. It has decompositions

ℙ_{m,n} = ⨁_{μ_{1},...,μ_{m}} ℙ_{m,n}^{𝔤𝔩_{m}}[(μ_{1}, …, μ_{m})] = ⨁_{ν_{1},...,ν_{n}} ℙ_{m,n}^{𝔤𝔩_{n}}[(ν_{1}, …, ν_{n})]

where the subspaces

ℙ_{m,n}^{𝔤𝔩_{m}}[(μ_{1}, …, μ_{m})] ≅ S^{μ_{1}}(ℂ^{n}) ⊗ ⋯ ⊗ S^{μ_{m}}(ℂ^{n}),
ℙ_{m,n}^{𝔤𝔩_{n}}[(ν_{1}, …, ν_{n})] ≅ S^{ν_{1}}(ℂ^{m}) ⊗ ⋯ ⊗ S^{ν_{n}}(ℂ^{m}),

are the weight subspaces with respect to the algebras 𝔤𝔩_{m} and 𝔤𝔩_{n} respectively.

In the explicit realization of ℙ_{m,n} as the polynomial algebra in mn variables ℂ[{x_{ij}}], 1 ≤ i ≤ m, 1 ≤ j ≤ n, the actions of 𝔤𝔩_{m}, 𝔤𝔩_{n} are given by

E_{ab}^{𝔤𝔩_{m}} = ∑_{i=1}^{n} x_{ai} \frac{∂}{∂x_{bi}} , a, b = 1, …, m,

E_{ij}^{𝔤𝔩_{n}} = ∑_{a=1}^{m} x_{ai} \frac{∂}{∂x_{aj}} , i, j = 1, …, n,

where E_{rs}^{𝔤𝔩_{k}} are the standard generators of the Lie algebra 𝔤𝔩_{k}, corresponding to the elementary matrices with 1 at the intersection of the r-th row and s-th column, and 0 elsewhere.

We have the following theorem, see [Zh, H].

Theorem 11. The 𝔤𝔩_{m} ⊗ 𝔤𝔩_{n} module ℙ_{m,n} has the decomposition

ℙ_{m,n} = ⨁_{Λ∈P_{min(m,n)}} V_{Λ}^{𝔤𝔩_{m}} ⊗ V_{Λ}^{𝔤𝔩_{n}}

where P_{j} denotes the collection of finite sequences Λ of nonnegative integers Λ_{1} ≥ ⋯ ≥ Λ_{j}, and V_{Λ}^{𝔤𝔩_{k}} for k ≥ j denotes the irreducible 𝔤𝔩_{k}-module with highest weight (Λ_{1}, …, Λ_{j}, 0, …, 0).

In particular, this theorem implies that a 𝔤𝔩_{n}-module V_{Λ}^{𝔤𝔩_{n}} may be realized as the subspace of 𝔤𝔩_{m}-singular vectors of weight Λ in ℙ_{m,n}.

We will use the special case of the (𝔤𝔩_{2}, 𝔤𝔩_{3}) duality, which gives the following decomposition of the polynomial algebra ℙ_{2,3}:

ℙ_{2,3} = ⨁_{m_{1}≥m_{2}≥0} V_{(m_{1},m_{2})}^{𝔤𝔩_{2}} ⊗ V_{(m_{1},m_{2},0)}^{𝔤𝔩_{3}}.

We identify a module V_{(m_{1},m_{2},0)}^{𝔤𝔩_{3}} with the the subspace of 𝔤𝔩_{2}-singular vectors in the algebra ℙ_{2,3} of weight (m_{1}, m_{2}) with respect to 𝔤𝔩_{2}:

V_{(m_{1},m_{2},0)}^{𝔤𝔩_{3}} ≅ ⨁_{l_{1}+l_{2}+l_{3}=m_{1}+m_{2}} (V_{(l_{1},0)}^{𝔤𝔩_{2}} ⊗ V_{(l_{2},0)}^{𝔤𝔩_{2}} ⊗ V_{(l_{3},0)}^{𝔤𝔩_{2}})^{sing} [(m_{1}, m_{2})].

Such identification gives the correspondence of the weight subspaces,

V_{(m_{1},m_{2},0)}^{𝔤𝔩_{3}}[(l_{1}, l_{2}, l_{3})] ≅ (V_{(l_{1},0)}^{𝔤𝔩_{2}} ⊗ V_{(l_{2},0)}^{𝔤𝔩_{2}} ⊗ V_{(l_{3},0)}^{𝔤𝔩_{2}})^{sing} [(m_{1}, m_{2})].

Our next goal is to describe a realization of V_{(l_{1},0)}^{𝔤𝔩_{2}} ⊗ V_{(l_{2},0)}^{𝔤𝔩_{2}} ⊗ V_{(l_{3},0)}^{𝔤𝔩_{2}} as a certain functional space. In the remaining part of this section l = (l_{1}, l_{2}, l_{3}) will denote a triple of non-negative integers.

To shorten formulas, we will omit superscripts for 𝔤𝔩_{2}-modules and operators, and write V_{(l,0)}, E_{ab} instead of V_{(l,0)}^{𝔤𝔩_{2}}, E_{ab}^{𝔤𝔩_{2}}, respectively. By a slight abuse of notation, we also write 1_{l} for the highest weight vector of the 𝔤𝔩_{2}-module V_{(l,0)}.

Let ℂ[z], ℂ(z) denote respectively the algebras of polynomial and rational functions in complex variables z = (z_{1}, z_{2}, z_{3}).

Let H_{k} denote the space of symmetric polynomials in variables t_{1}, …, t_{k} of degree at most 2 in each t_{1}, …, t_{k}, with coefficients in ℂ[z]. We also set

H = ⨁_{k∈ℤ_{≥0}} H_{k} .

For any function φ of t_{1}, …, t_{k}, we denote

Sym(φ(t_{1}, …, t_{k})) = ∑_{σ∈S_{k}} φ(t_{σ(1)}, …, t_{σ(k)})

Proposition 12. [TV3] The space H has a structure of a 𝔤𝔩_{2}-module, depending on l, such that the action of generators {E_{ab}} on a function φ ∈ H_{k} is given by

(E_{11}φ)(t_{1}, …, t_{k}) = (l_{1} + l_{2} + l_{3} − k)φ(t_{1}, …, t_{k}),

(E_{22}φ)(t_{1}, …, t_{k}) = kφ(t_{1}, …, t_{k}),

(E_{12}φ)(t_{1}, …, t_{k−1}) = lim_{t_{k}→∞} \frac{φ(t_{1}, …, t_{k})}{t_{k}^{2}},

\begin{matrix}(E_{21}φ)(t_{1}, …, t_{k+1}) = \frac{1}{k!} Sym (φ(t_{1}, …, t_{k}) (∏_{a=1}^{3}(t_{k+1} − z_{a} + l_{a}) ∏_{i=1}^{k} \frac{t_{k+1} − t_{i} − 1}{t_{k+1} − t_{i}} − \\ ∏_{a=1}^{3}(t_{k+1} − z_{a}) ∏_{i=1}^{k} \frac{t_{k+1} − t_{i} + 1}{t_{k+1} − t_{i}})) . \end{matrix}

One can easily check that multiplication by any function of z commutes with the 𝔤𝔩_{2} action above; thus, H can be regarded as a 𝔤𝔩_{2} ⊗ ℂ[z] - module.

Proposition 13. [TV3] The linear map

ϕ[l] : V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)} → H ,

ϕ[l] (E_{21}^{k_{1}}1_{l_{1}} ⊗ E_{21}^{k_{2}}1_{l_{2}} ⊗ E_{21}^{k_{3}}1_{l_{3}}) \frac{l_{1}!l_{2}!l_{3}!}{(l_{1} − k_{1})!(l_{2} − k_{2})!(l_{3} − k_{3})!} =

Sym (∏_{a=1}^{3} ( ∏_{i=1}^{k_{1}+···+k_{a−1}}(t_{i} − z_{a}) ∏_{j=k_{1}+···+k_{a}+1}^{k_{1}+k_{2}+k_{3}}(t_{j} − z_{a} + l_{a})) ∏_{i<j} \frac{t_{i} − t_{j} + 1}{t_{i} − t_{j}}) , (6.1)

is a homomorphism of 𝔤𝔩_{2}-modules.

Let H_{k}[l] denote the subspace of H_{k}, consisting of functions φ ∈ H_{k}, satisfying additional ”admissibility” conditions

φ(z_{i}, z_{i} − 1, …, z_{i} − l_{i}, t_{l_{i}+2}, …, t_{k}) ≡ 0,  (6.2)

imposed if k > l_{i}, i = 1, 2, 3. We also denote

H[l] = ⨁_{k∈ℤ_{≥0}} H_{k}[l] .

One can check that H[l] is a 𝔤𝔩_{2} ⊗ ℂ[z]-submodule of H.

Proposition 14. 1. The map ϕ[l] is injective, and

Im ϕ[l] ⊗_{ℂ[z]} ℂ(z) = H[l] ⊗_{ℂ[z]} ℂ(z).  (6.3)

2. There exists a homomorphism of 𝔤𝔩_{2} ⊗ ℂ[z]-modules

I[l] : H[l] → (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)}) ⊗ ℂ(z),

such that

(I[l] ∘ ϕ[l]) v = v,  for any v ∈ V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)}.  (6.4)

Moreover, for any φ ∈ H[l], the vector I[l]φ, rationally depending on z, may have at most simple poles, located at the hyperplanes

z_{j} − r = z_{i} − l_{i}, r = 0, 1, …, min(l_{i}, l_{j}) − 1, 1 ≤ i ≤ j ≤ 3.

The Proposition is proved in Appendix B.

Introduce X [l] ∈ ℂ[z] by

X [l; z] = ∏_{r=1}^{l_{2}}(z_{1} − l_{1} − z_{2} + r) ∏_{r=1}^{l_{3}}(z_{1} − l_{1} − z_{3} + r) ∏_{r=1}^{l_{3}}(z_{2} − l_{2} − z_{3} + r),

and set

N [l] = X [l] I[l].  (6.5)

Corollary 15. The formula (6.5) defines a homomorphism of 𝔤𝔩_{2} ⊗ ℂ[z]-modules

N [l] : H[l] → (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)}) ⊗ ℂ[z].

Proof. For any φ ∈ H[l], the possible simple poles of I[l]φ are offset by X [l]. Therefore, N [l]φ is polynomial in z. The 𝔤𝔩_{2}-intertwining property for N [l] follows from the fact that I[l] is a 𝔤𝔩_{2}-homomorphism, and the commutativity of multiplication by any element from ℂ[z] with the action of 𝔤𝔩_{2}.

Now we let formal variables z take particular complex values; specifying z = z_{0} for some z_{0} ∈ ℂ^{3} will be reflected by adding z_{0} to the notation. For example, ϕ[l; z_{0}] will denote the composition of ϕ[l] with the homomorphism ℂ[z] → ℂ of evaluation at z = z_{0}, and H_{k}[l; z_{0}] will consist of symmetric polynomials in t_{1}, …, t_{k} with complex coefficients, obtained from elements of H_{k}[l] by specializing z = z_{0}.

We now establish certain properties of operators ϕ[l; z_{0}], N [l; z_{0}] for special values of z_{0}. Recall the following standard decompositions of the tensor product of 𝔤𝔩_{2}-modules:

V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)} ≅ ( ⨁_{i=0}^{min(l_{1},l_{2})} V_{(l_{1}+l_{2}−i,i)}) ⊗ V_{(l_{3},0)} ≅ V_{(l_{1},0)} ⊗ ( ⨁_{i=0}^{min(l_{2},l_{3})} V_{(l_{2}+l_{3}−i,i)}) .

Proposition 16. Let z_{0} ∈ ℂ^{3}.

1. If z_{0} is generic from the hyperplane z_{1} − z_{2} − l_{1} + s = 0, 0 ≤ s < min(l_{1}, l_{2}), then

ker ϕ[l; z_{0}] = ( ⨁_{i=s+1}^{min(l_{1},l_{2})} V_{(l_{1}+l_{2}−i,i)}) ⊗ V_{(l_{3},0)} .

2. If z_{0} is generic from the hyperplane z_{2} − z_{3} − l_{2} + s = 0, 0 ≤ s < min(l_{2}, l_{3}), then

ker ϕ[l; z_{0}] = V_{(l_{1},0)} ⊗ ( ⨁_{i=s+1}^{min(l_{2},l_{3})} V_{(l_{2}+l_{3}−i,i)}) ,

This Proposition will be proved in Appendix B. As a consequence, we obtain the following properties of operators N [l; z_{0}].

Corollary 17. Let z_{0} ∈ ℂ^{3}.

1. If z_{0} is generic from the hyperplane z_{1} − l_{1} − z_{2} + s = 0, 0 ≤ s < min(l_{1}, l_{2}), then

im N [l; z_{0}] ⊂ ( ⨁_{i=s+1}^{min(l_{1},l_{2})} V_{(l_{1}+l_{2}−i,i)}) ⊗ V_{(l_{3},0)}

2. If z_{0} is generic from the hyperplane z_{2} − l_{2} − z_{3} + s = 0, 0 ≤ s < min(l_{2}, l_{3}), then

im N [l; z_{0}] ⊂ V_{(l_{1},0)} ⊗ ( ⨁_{i=s+1}^{min(l_{2},l_{3})} V_{(l_{2}+l_{3}−i,i)})

Proof. Let φ ∈ H[l; z_{0}]. Then it follows from the definitions and (6.3) that

ϕ[l; z_{0}] ∘ N [l; z_{0}]φ = X [l; z_{0}]φ .  (6.6)

For any point z_{0} from the hyperplane z_{1} − l_{1} − z_{2} + s = 0 with 0 ≤ s < min(l_{1}, l_{2}), we have that X [l; z_{0}] = 0, and therefore N [l; z_{0}]φ ⊂ ker ϕ[l; z_{0}]. Hence, the first assertion follows from Proposition 16.

The proof of the second assertion is similar.

Note that admissibility conditions (6.2) are invariant under simultaneous permutations of triples l and z, i.e. we have

H[l; z] = H[σ(l); σ(z)], σ ∈ S_{3}.

Proposition 18. There exists a family of linear operators Ř_{l,l′}(u) : V_{(l,0)} ⊗ V_{(l′,0)} → V_{(l′,0)} ⊗ V_{(l,0)}, rationally depending on a complex parameter u, such that

Ř_{l,l′}(u)1_{l} ⊗ 1_{l′} = \frac{∏_{j=1}^{l′}(u − l + j)}{∏_{j=1}^{l}(−u − l′ + j)}1_{l′} ⊗ 1_{l},  (6.7)

and the following diagram is commutative for generic z:

H[s_{1}(l); s_{1}(z)] ⟶^{N [s_{1}(l);s_{1}(z)]} V_{(l_{2},0)} ⊗ V_{(l_{1},0)} ⊗ V_{(l_{3},0)} ∥  ↑^{Ř_{l_{1}l_{2}}(z_{1}−z_{2})⊗1} H[l; z]  ⟶^{N [l;z]} V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)} ∥  ↓^{1⊗Ř_{l_{2}l_{3}}(z_{2}−z_{3})} H[s_{2}(l); s_{2}(z)] ⟶^{N [s_{2}(l);s_{2}(z)]} V_{(l_{1},0)} ⊗ V_{(l_{3},0)} ⊗ V_{(l_{2},0)} .  (6.8)

Proposition 18 is proved in Appendix B, where it is shown that the operators Ř_{l,l′}(u) are the standard rational R-matrices, renormalized by (6.7).

7. The main construction

Let U be an irreducible finite-dimensional 𝔰𝔩_{3}-module with highest weight Λ. Set

m = ⟨α_{1} + 2α_{2}, Λ⟩, k = ⟨α_{2}, Λ⟩.

The 𝔤𝔩_{3}-module V_{(m−k,k,0)}^{𝔤𝔩_{3}}, regarded by restriction as an 𝔰𝔩_{3}-module, is isomorphic to U. We fix an identification (which is unique up to a scalar)

Θ : U → ⨁_{l∈ℤ_{≥0}^{3}} (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)],

so that Θ maps a weight subspace U[μ] to (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)], where the corresponding l = (l_{1}, l_{2}, l_{3}) ∈ ℤ_{≥0}^{3} is determined by

l_{1} − l_{2} = ⟨α_{1}, μ⟩,  l_{2} − l_{3} = ⟨α_{2}, μ⟩,  l_{1} + l_{2} + l_{3} = m.  (7.1)

Under this identification the Weyl group for 𝔰𝔩_{3} coincides with the symmetric group S_{3} naturally acting on ℤ_{≥0}^{3}.

For any l ∈ ℤ_{≥0}^{3} we fix a basis {v_{i}[l]} of the space (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)]. We assume that v_{i}[w(l)] = w̄v_{i}[l], where the operators w̄ correspond to the action of the Weyl group given by Lemma 2. Note that if w(l) = l, then the restriction of w̄ on (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)] is the identity operator.

We have the following Lemma, proved in Appendix C.

Lemma 19. For any l = (l_{1}, l_{2}, l_{3}) ∈ ℤ_{≥0}^{3}, and k ∈ ℤ_{≥0}, we have

dim H_{k}[l; z]^{sing} = dim (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)] ,

and there exists a ℂ[z]-basis {ς_{i}[l]} of H_{k}[l]^{sing}, such that

ς_{i}[l; z] = ς_{i}[σ(l); σ(z)],  σ ∈ S_{3}.  (7.2)

Define the operators Υ[l] : (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)] → H[l]^{sing} by

Υ[l]v_{i}[l] = ς_{i}[l].

By construction, the operators Υ[l] satisfy

(V_{(l_{2},0)} ⊗ V_{(l_{1},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)] ⟶^{Υ[s_{1}(l);s_{1}(z)]} H_{k}[s_{1}(l); s_{1}(z)]^{sing} s̄_{1}↑  ∥ (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)] ⟶^{Υ[l;z]}  H_{k}[l; z]^{sing} s̄_{2}↓  ∥ (V_{(l_{1},0)} ⊗ V_{(l_{3},0)} ⊗ V_{(l_{2},0)})^{sing} [(m − k, k)] ⟶^{Υ[s_{2}(l);s_{2}(z)]} H_{k}[s_{2}(l); s_{2}(z)]^{sing}.  (7.3)

We now introduce our main object — the operators N_{μ}(λ) ∈ End_{ℂ}(U[μ]) — by setting

N_{μ}(λ) = Θ^{−1} ∘ N [l; z] ∘ Υ[l; z] ∘ Θ .  (7.4)

Here μ and l_{1}, l_{2}, l_{3} are as in (7.1), and z_{1}, z_{2}, z_{3} satisfy

(z_{1} − l_{1}) − (z_{2} − l_{2}) = ⟨α_{1}, λ + ρ⟩,  (z_{2} − l_{2}) − (z_{3} − l_{3}) = ⟨α_{2}, λ + ρ⟩ .  (7.5)

Theorem 20. The operator N(λ) = ⨁_{μ} N_{μ}(λ) is a regularizing operator for the 𝔤-module U. The corresponding Weyl group representation w ↦ a_{w} is such that a_{s_{i}} = s̄_{i}.

Proof. Corollary 15 implies that the operator N(λ) depends on λ ∈ 𝔥^{∗} polynomially.

To check (5.2), we use the result of [TV2], where it was shown that the action of the dynamic Weyl group operators A_{w} in ℙ_{2,3} coincides with the action of operators Ř. More precisely, we have the following commutative diagram:

U[s_{1}μ] ⟶ (V_{(l_{2},0)} ⊗ V_{(l_{1},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)] ^{A_{s_{1}}^{μ}(λ)}↑  ↑^{Ř_{l_{1}l_{2}}(z_{1}−z_{2})⊗1} U[μ] ⟶ (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(m − k, k)] ^{A_{s_{2}}^{μ}(λ)}↓  ↓^{1⊗Ř_{l_{2}l_{3}}(z_{2}−z_{3})} U[s_{2}μ] ⟶ (V_{(l_{1},0)} ⊗ V_{(l_{3},0)} ⊗ V_{(l_{2},0)})^{sing} [(m − k, k)] .  (7.6)

Combining (7.6) with (6.8) and (7.3), we get the commutative diagram

\begin{matrix}U[s_{1}μ] & ⟶^{N_{s_{1}μ}(s_{1}·λ)} & U[s_{1}μ] \\ s̄_{1}↑ & & ↑^{A_{s_{1}}^{μ}(λ)} \\ U[μ] & ⟶^{N_{μ}(λ)} & U[μ] \\ s̄_{2}↓ & & ↓^{A_{s_{2}}^{μ}(λ)} \\ U[s_{2}μ] & ⟶^{N_{s_{2}μ}(s_{2}·λ)} & U[s_{2}μ] \end{matrix} ,

which implies (5.2).

Next, the operator N [l; z]^{−1} = X [l; z]^{−1}ϕ[l; z] is singular only at the hyperplanes

z_{1} − l_{1} − z_{2} + s = 0, 0 ≤ s < l_{2} ,
z_{2} − l_{2} − z_{3} + s = 0, 0 ≤ s < l_{3} ,
z_{1} − l_{1} − z_{3} + s = 0, 0 ≤ s < l_{3} ,

which correspond respectively to the hyperplanes

χ_{α_{1},l_{2}−s}(λ) = 0 , 0 ≤ s < l_{2} ,
χ_{α_{2},l_{3}−s}(λ) = 0 , 0 ≤ s < l_{3} ,
χ_{α_{1}+α_{2},l_{3}−s}(λ) = 0 0 ≤ s < l_{3} .

Now it follows from Lemma 6 that N(λ) satisfies (5.1).

Finally, let us show that for any u ∈ U[μ], the vector Ξ_{N}(λ)u ∈ U(𝔫^{−}) ⊗ U polynomially depends on λ. According to (3.1), this is equivalent to regularity of the sum ∑_{j} (S_{λ}^{−1})_{ij} ω(g_{j})N(λ)u for any i at any hyperplane χ_{α,r}(λ) = 0, α ∈ Δ^{+}, r = 1, 2, . . ..

Consider the case α = α_{1}.

Let {g_{j}} be the Poincare-Birkhoff-Witt basis of U(𝔫^{−}), corresponding to an ordering of Δ^{+} with α_{1} being the last root. It is known (see [ES]) that (S_{λ}^{−1})_{ij} is regular at the hyperplane χ_{α_{1},r}(λ) = 0 unless g_{j} = g̃_{j}f_{1}^{r} for some g̃_{j} ∈ U(𝔫^{−}). In the last case (S_{λ}^{−1})_{ij} may have a simple pole there, and the desired regularity will follow if ω(g̃_{j})e_{1}^{r}N(λ)u vanishes at the hyperplane χ_{α_{1},r}(λ) = 0.

We prove that e_{1}^{r}N(λ)u = 0 at the hyperplane χ_{α_{1},r}(λ) = 0 using the explicit realization Θ of the module U in ℙ_{2,3}. The operator e_{1}^{r} : U[μ] → U[μ + rα_{1}] corresponds to the operator

(E_{12}^{𝔤𝔩_{3}})^{r} : (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(m−k, k)] → (V_{(l_{1}+r,0)} ⊗ V_{(l_{2}−r,0)} ⊗ V_{(l_{3},0)})^{sing} [(m−k, k)] , if r ≤ l_{2}, and is equal to zero, if r > l_{2}. There is nothing to prove in the latter case. The claim in the former case is implied by the following lemma.

Lemma 21. Assume that r ≤ l_{2}. Let z_{0} be a generic point from the hyperplane (z_{1} − l_{1}) − z_{2} + (l_{2} − r) = 0. Then the following composition of operators vanishes:

H[l; z_{0}] ⟶^{N [l;z_{0}]} V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)} ⟶^{(E_{12}^{𝔤𝔩_{3}})^{r}} V_{(l_{1}+r,0)} ⊗ V_{(l_{2}−r,0)} ⊗ V_{(l_{3},0)}.  (7.7)

Proof. Since r ≤ l_{2}, the product X [l; z] contains the factor z_{1}−l_{1}−z_{2}+(l_{2}−r), corresponding to χ_{α_{1},r}(λ), and X [l; z_{0}] = 0 If l_{2} − r > min(l_{1}, l_{2}), then I[l]φ is regular at z = z_{0} for any φ ∈ H[l], and the operator N [l; z_{0}] = X [l; z_{0}] I[l; z_{0}] vanishes.

If l_{2} − r ≤ min(l_{1}, l_{2}), then by Proposition 17, the image of N [l; z_{0}] is contained in (⨁_{i=l_{2}−r+1}^{min(l_{1},l_{2})} V_{(l_{1}+l_{2}−i,i)}) ⊗ V_{(l_{3},0)}. Since (E_{12}^{𝔤𝔩_{3}})^{r} commutes with the 𝔤𝔩_{2} action and

V_{(l_{1}+r,0)} ⊗ V_{(l_{2}−r,0)} ⊗ V_{(l_{3},0)} ≅ (⨁_{i=0}^{l_{2}−r} V_{(l_{1}+l_{2}−i,i)}) ⊗ V_{(l_{3},0)} ,

the composition (7.7) vanishes.

The case α = α_{2} can be considered similarly to the case α = α_{1}.

Let now α = α_{1} + α_{2}. In this case we have α = s_{2}α_{1}, and therefore

χ_{α,r}(λ) = ⟨α, λ + ρ⟩ − r = ⟨s_{2}α_{1}, λ + ρ⟩ − r = ⟨α_{1}, s_{2} · λ + ρ⟩ − r = χ_{α_{1},r}(s_{2} · λ).

Recall that Ξ_{N}(λ)u is regular at the hyperplane χ_{α,r}(λ) = 0 if and only if the operator Φ_{λ}^{N(λ)u} is regular at the same hyperplane.

Consider λ from the hyperplane ⟨α_{2}, λ + ρ⟩ = −n, for a large enough nonnegative integer n. Then M_{λ} is contained as a proper submodule in M_{s_{2}·λ}, and we have

Φ_{λ}^{N(λ)u} = Φ_{s_{2}·λ}^{A(s_{2}·λ)N(s_{2}·λ)u}|_{M_{λ}} .

Since the operator Φ_{s_{2}·λ}^{N(s_{i}·λ)u} is regular at the hyperplanes χ_{α_{1},r}(s_{2} · λ) = 0, so is the operator Φ_{λ}^{N(λ)u}. Hence Ξ_{N}(λ)u has no pole at the hyperplane χ_{α,r}(λ) = χ_{α_{1},r}(s_{2} · λ) = 0, provided ⟨α_{2}, λ + ρ⟩ = −n.

The rational function χ_{α,r}(λ)Ξ_{N}(λ)u is regular at the hyperplane χ_{α,r}(λ) = 0, and vanishes at infinitely many points of the hyperplane. Therefore, the function must vanish at the hyperplane identically, that is Ξ_{N}(λ)u has no pole at the hyperplane χ_{α,r}(λ) = 0.

8. Resonance conditions

Let 𝔤 be a semisimple Lie algebra, and let U be an irreducible finite-dimensional 𝔤-module with nontrivial zero weight subspace U[0].

Let N(λ) ∈ End_{ℂ}(U) be a regularizing operator. Introduce an End_{ℂ}(U[0])-valued function Ψ(λ, x), where λ ∈ 𝔥^{∗}, x ∈ 𝔥, by

Ψ(λ, x)u = Tr |_{M_{λ}} (Φ_{λ}^{N(λ)u}e^{x}) .

Special cases of the function Ψ(λ, x) were used in [ES] to give a representation-theoretic proof of the algebraic integrability of Calogero-Sutherland systems. In [EV] it was shown that the trace functions satisfy remarkable differential and difference equations.

An important property of the function Ψ(λ, x) is that it satisfies certain resonance conditions.

Proposition 22. Let N = N(λ) be a regularizing operator acting in a finite-dimensional 𝔤-module U with U[0] ≠ 0, and let Ψ(λ, x) be the corresponding trace function.

Let α ∈ Δ^{+} and k ∈ ℤ_{≥0} be such that U[kα] ≠ 0. Then for any λ ∈ 𝔥^{∗} such that χ_{α,k}(λ) = 0, and any u ∈ ker N(λ) ⊂ U[0], we have

Ψ(λ, x)u = Ψ(s_{α} · λ, x)u .

Proof. Let λ be generic from the hyperplane χ_{α,k}(λ) = 0. Then M_{λ} contains a unique proper submodule M_{s_{α}·λ}, and N(λ)u = 0 implies that the image of Φ_{λ}^{N(λ)u} is contained in M_{s_{α}·λ} ⊗U. Therefore, the trace of the operator Φ_{λ}^{N(λ)u} in M_{λ} is equal to the trace of its resrtiction to M_{s_{α}·λ}. Moreover, the restriction of Φ_{λ}^{N(λ)u} to the submodule M_{s_{α}·λ} coincides with Φ_{s_{α}·λ}^{N(s_{α}·λ)u}, and we have

Ψ(λ, x)u = Tr |_{M_{λ}} (Φ_{λ}^{N(λ)u}e^{x}) = Tr |_{M_{sα·λ}} (Φ_{λ}^{N(λ)u}e^{x}) =
Tr |_{M_{sα·λ}} (Φ_{s_{α}·λ}^{N(s_{α}·λ)u}e^{x}) = Ψ(s_{α} · λ, x)u .

Example: Let 𝔤 = 𝔰𝔩_{2}. We identify 𝔥^{∗} with ℂ by associating 𝔥^{∗} ∋ λ ↔ ⟨α, λ⟩ ∈ ℂ.

Let U be an irreducible finite-dimensional 𝔰𝔩_{2}-module with even highest weight Λ. In this case the weight subspace U[0] is one-dimensional, End(U[0]) ≅ ℂ, and the function Ψ(λ, x) is scalar-valued. It satisfies the resonance conditions

Ψ(λ, x) = Ψ(s · λ, x), λ = 0, 1, 2, …, Λ/2 − 1 .

Appendix A. Formal monomials

To prove Theorem 4 we use the calculus of formal monomials in U(𝔫^{−}), developed in [FFM].

Lemma 23. Let f ∈ 𝔫^{−}, X ∈ 𝔤. For any n ∈ ℤ_{≥0} we have the following identity in U(𝔫^{−}),

Xf^{n} = ∑_{k=0}^{∞} \binom{n}{k}f^{n−k} \underbrace{[[[X, f], …,], f]}_{k times} .  (A.1)

This lemma is proved by induction. Note that the summation over k is finite.

Introduce an associative algebra 𝔸, with unit 1 and generators f_{i}^{γ}, i = 1, …, dim 𝔥, γ ∈ ℂ, subject to the relations

\underbrace{[f_{i}, [, …, [f_{i}, f_{j}] …]]}_{a_{ij} times} = 0,  a_{ij} = \frac{2⟨α_{i}, α_{j}⟩}{⟨α_{i}, α_{i}⟩},  (A.2)

f_{i}^{β}f_{i}^{γ} = f_{i}^{β+γ},  β, γ ∈ ℂ,  (A.3)
f_{i}f_{j}^{γ} = ∑_{k=0}^{∞} \binom{γ}{k}f_{j}^{γ−k} \underbrace{[[… [f_{i}, f_{j}], …,], f_{j}]}_{k times},  γ ∈ ℂ.  (A.4)

where the commutator [·, ·] is defined as [X, Y] = XY − Y X.

Lemma 24. There exists an algebra homomorphism ι : U(𝔫^{−}) → 𝔸 such that ι(f_{i}) = f_{i} for any i = 1, …, dim 𝔥. The homomorphism ι is injective.

Proof. It is known that U(𝔫^{−}) is the quotient of the free Lie algebra with generators {f_{i}}, by the ideal, generated by the Serre relations (A.2). Therefore, ι extends from the generators to the entire algebra U(𝔫^{−}).

Next, we note that the algebra 𝔸 is nonzero, i.e. the ideal, generated by the relations (A.4) is proper. Indeed, a linear functional φ on 𝔸, defined on monomials by

φ(f_{i_{1}}^{γ_{1}} … f_{i_{l}}^{γ_{l}}) = 1,

is well-defined, and is nonzero, so 𝔸 ≠ 0.

Let λ ∈ 𝔥^{∗} be generic. Introduce a 𝔤-module M̃_{λ}, which is equal to 𝔸 as a vector space, with f_{i} acting naturally on the left, h ∈ 𝔥 acting by

hf_{i_{1}}^{γ_{1}} … f_{i_{l}}^{γ_{l}} = ⟨h, λ − ∑_{k=1}^{l} γ_{k}α_{i_{k}}⟩f_{i_{1}}^{γ_{1}} … f_{i_{l}}^{γ_{l}}.

The action of operators e_{i} is determined by the condition e_{i}1 = 0, and the commutation formula

e_{i}f_{j}^{γ} = ∑_{k=0}^{∞} \binom{γ}{k}f_{j}^{γ−k} \underbrace{[[… [e_{i}, f_{j}], …,], f_{j}]}_{k times}, γ ∈ ℂ.

The module M̃_{λ} is an obvious analog of the Verma module M_{λ}.

For any X, Y ∈ U(𝔫^{−}), consider the action of ω(Y) ∈ U(𝔤) on the element ι(X)1 ∈ M̃_{λ}. We have ω(Y)ι(X)1 = S_{λ}(X, Y)1, and therefore ι(X) = 0 implies S_{λ}(X, Y) = 0 for any Y ∈ U(𝔫^{−}). Since the Shapovalov form S_{λ} is nondegenerate for generic λ, this means that X = 0. Thus, ι is an injection.

We identify U(𝔫^{−}) with its image ι(U(𝔫^{−})) ⊂ 𝔸. We say that an element X ∈ 𝔸 makes sense, if X ∈ U(𝔫^{−}).

Example 1. Let 𝔤 = 𝔰𝔩_{3}, and let γ_{1}, γ_{2}, γ_{3} ∈ ℂ. If γ_{1} + γ_{3}, γ_{2} ∈ ℤ_{≥0}, and γ_{1} + γ_{3} ≥ γ_{2}, then the monomial f_{1}^{γ_{1}}f_{2}^{γ_{2}}f_{1}^{γ_{3}} makes sense, and we can write

f_{1}^{γ_{1}}f_{2}^{γ_{2}}f_{1}^{γ_{3}} = ∑_{j=0}^{∞} \binom{γ_{2}}{j}\binom{γ_{3}}{j}f_{1}^{γ_{1}+γ_{3}−j}f_{2}^{γ_{2}−j}[f_{2}, f_{1}]^{j}.

Consider the completion 𝔸 ˆ⊗ U(𝔫^{−}), consisting of possibly infinite sums ∑_{i} X_{i} ⊗ Y_{i} with X_{i} ∈ 𝔸 and homogeneous Y_{i} ∈ U(𝔫^{−}), such that wt(Y_{i}) → ∞ when i → ∞.

We following two Lemmas can be easily verified; we give sketches of proof, and leave more technical details to a reader.

Lemma 25. There exists an algebra homomorphism

Δ : 𝔸 → 𝔸 ˆ⊗ U(𝔫^{−}),

such that

Δ(f_{i}^{γ}) = ∑_{j=0}^{∞} \binom{γ}{j}f_{i}^{γ−j} ⊗ f_{i}^{j}.

Sketch of proof. We have to check that the comultiplication above can be extended to an algebra homomorphism, i.e. that it respects the defining relations (A.4). For each such relation of homogeneous weight γ, it amounts to verifying a family of identities in weight subspaces 𝔸[ν] ⊗ U(𝔫^{−})[γ − ν]. These identities are are valid for any nonnegative integer γ, because in this case they are equivalent to the consistency of the comultiplication in U(𝔫^{−}). Since these identities are polynomial in γ, they are satisfied for arbitrary γ.

It is easy to see that if X ∈ U(𝔫^{−}) ⊂ 𝔸, then Δ(X) ∈ U(𝔫^{−}) ⊗ U(𝔫^{−}) ⊂ 𝔸 ˆ⊗ U(𝔫^{−}).

Lemma 26. Let J ∈ ℤ_{≥0}, γ_{1}, …, γ_{l} ∈ ℂ. Let {g_{i}} be a homogeneous basis of U(𝔫^{−}). Then there exist J_{1}, …, J_{l} ∈ ℤ_{≥0}, such that any monomial f_{i_{1}}^{γ_{1}−j_{1}} … f_{i_{l}}^{γ_{l}−j_{l}} with j_{1}, …, j_{l} ≤ J, can be written as

f_{i_{1}}^{γ_{1}−j_{1}} … f_{i_{l}}^{γ_{l}−j_{l}} = f_{i_{1}}^{γ_{1}−J_{1}} … f_{i_{l}}^{γ_{l}−J_{l}} ∑_{i} q_{i}(γ_{1}, …, γ_{l})g_{i},

for some polynomials q_{i}(γ_{1}, …, γ_{l}).

Sketch of proof. The statement is obvious for l = 1. Using induction on l, we can write

f_{i_{1}}^{γ_{1}−j_{1}} … f_{i_{l−1}}^{γ_{l−1}−j_{l−1}}f_{i_{l}}^{γ_{l}−j_{l}} = f_{i_{1}}^{γ_{1}−J_{1}} … f_{i_{l−1}}^{γ_{l−1}−J_{l−1}} (∑_{i} q_{i}(γ_{1}, …, γ_{l})g_{i}) f_{i_{l}}^{γ_{l}−j_{l}}.

Then we can move all f_{i_{l}} to the left by commuting them with {g_{i}}; nilpotency of the adjoint action of f_{i_{l}} will guarantee that all only finite number of f_{i_{l}}’s can be absorbed.

We now use these resulst to give a proof of Theorem 4.

Proof of Theorem 4. We represent the left side of (4.4) as

Φ_{λ}^{u}v_{w·(λ+μ)} = Φ_{λ}^{u}F_{w}(λ + μ)1_{λ+μ} = Δ(F_{w}(λ + μ))Ξ(λ)(1_{λ} ⊗ u) =

∑_{j_{1},...,j_{l}∈ℤ_{≥0}} ∑_{i,j} q_{j_{1},... ,j_{l}}(γ_{1}(λ + μ), …, γ_{l}(λ + μ)) ×

f_{i_{1}}^{γ_{1}(λ+μ)−j_{1}} … f_{i_{l}}^{γ_{l}(λ+μ)−j_{l}}g_{i}1_{λ} ⊗ (S_{λ}^{−1})_{ij} f_{i_{1}}^{j_{1}} … f_{i_{l}}^{j_{l}}ω(g_{j})u.  (A.5)

Here the coefficients q_{j_{1},... ,j_{l}}(γ_{1}(λ + μ), …, γ_{l}(λ + μ)) are certain polynomials in γ_{1}(λ + μ), …, γ_{l}(λ + μ). The summation in (A.5) is actually finite, because the element f_{i_{1}}^{j_{1}} … f_{i_{l}}^{j_{l}} acts as zero in U if any of j_{1}, …, j_{l} is big enough.

Next, we have that γ_{k}(λ + μ) = γ_{k}(λ) + ⟨β_{k}, μ⟩ for some β_{k} ∈ Δ^{+}. Hence, for all nonzero monomials in the sum, we have

γ_{k}(λ + μ) − j_{k} = γ_{k}(λ) + ⟨β_{k}, μ⟩ − j_{k} ≥ γ_{k}(λ) − J, k = 1, …, l,

for some J ∈ ℤ_{≥0}. Lemma 26 implies that there exist J_{1}, …, J_{l} ∈ ℤ_{≥0}, such that we can write

Δ(F_{w}(λ + μ))Ξ(λ) = (f_{i_{1}}^{γ_{1}(λ)−J_{1}} … f_{i_{l}}^{γ_{l}(λ)−J_{l}} ⊗ 1) (∑_{i} g_{i} ⊗ P_{i}(λ)) ,  (A.6)

for a certain finite collection of End_{ℂ}(U)-valued rational functions P_{i}(λ).

We now make two claims, which imply Theorem 4.

First, in the summation formula (A.6) we have

P_{i}(λ) ≡ 0,  if wt(g_{i}) ∉ wt(f_{i_{1}}^{J_{1}} … f_{i_{l}}^{J_{l}}) − Q^{+}.  (A.7)

Second, for some rational End_{ℂ}(U)-valued functions A_{w}(λ), we have

(f_{i_{1}}^{γ_{1}(λ)−J_{1}} … f_{i_{l}}^{γ_{l}(λ)−J_{l}} ⊗ 1) (∑_{i}′g_{i} ⊗ P_{i}(λ)) = F_{w}(λ) ⊗ A_{w}(λ),  (A.8)

where _{∑i}′ denotes the sum of terms with wt(g_{i}) = wt(f_{i_{1}}^{J_{1}} … f_{i_{l}}^{J_{l}}).

To verify these claims, we consider a dominant integral weight λ such that λ ≫ 0. One can check that for any u ∈ U we have

Δ(F_{w}(λ + μ))Ξ(λ)(1_{λ} ⊗ u) = v_{w·λ} ⊗ ũ + lower order terms ,  (A.9)

for some ũ ∈ U. In particular, the term g_{i} ⊗ P_{i}(λ) is nonzero only if

wt(f_{i_{1}}^{γ_{1}(λ)−J_{1}} … f_{i_{l}}^{γ_{l}(λ)−J_{l}}) + wt(g_{i}) + λ ≤ w · λ.  (A.10)

From the definition of γ_{k}(λ) it follows that

λ + wt(f_{i_{1}}^{γ_{1}(λ)} … f_{i_{l}}^{γ_{l}(λ)}) = w · λ,

and thus (A.9) implies (A.7) for the given λ. Since the rational functions P_{i}(λ) vanish for all sufficiently large dominant integral weights, they must vanish for all λ.

Using equation (A.8), we define the dynamical Weyl group operator A_{w}(λ) for integral dominant λ ≫ 0 by the rule A_{w}(λ)u = ũ. We now establish that A_{w}(λ) rationally depends on λ, and verify our second claim.

By Lemma 26, we can write

F_{w}(λ)f_{i_{1}}^{γ_{1}(λ)−J_{1}} … f_{i_{l}}^{γ_{l}(λ)−J_{l}} ∑_{i} Q_{i}(λ)g_{i},

for some polynomials Q_{i}(λ) in λ ∈ 𝔥^{∗}. Equation (A.9) then becomes

(f_{i_{1}}^{γ_{1}(λ)−J_{1}} … f_{i_{l}}^{γ_{l}(λ)−J_{l}} ⊗ 1) (∑_{i}′g_{i}1 ⊗ P_{i}(λ)u) =

(f_{i_{1}}^{γ_{1}(λ)−J_{1}} … f_{i_{l}}^{γ_{l}(λ)−J_{l}} ⊗ 1) (∑_{i}′Q_{i}(λ)g_{i}1 ⊗ A_{w}(λ)u) . (A.11)

or equivalently

∑_{i}′g_{i}1 ⊗ P_{i}(λ)u = ∑_{i}′Q_{i}(λ)g_{i}1 ⊗ A_{w}(λ)u.

It is valid for dominant integral λ ≫ 0, and implies that for all i we have

A_{w}(λ) = \frac{P_{i}(λ)}{Q_{i}(λ)},  if wt(g_{i}) = wt(f_{i_{1}}^{J_{1}} … f_{i_{l}}^{J_{l}}).  (A.12)

Therefore, A_{w}(λ) can be extended to a rational End_{ℂ}(U)-valued function, such that (A.12) is satisfied for all λ. Equations (A.11), (A.8), and (4.4) follow for all λ.

Appendix B. Yangians, weight functions and R-matrices

We recall some facts about the Yangian Y (𝔤𝔩_{2}), and explain how to get weight functions as matrix elements of evaluation Yangian modules. For more details on Yangians, see for example [CP].

The Yangian Y (𝔤𝔩_{2}) is a Hopf algebra with generators T_{ij}^{(k)}, i, j = 1, 2; k = 1, 2, 3, . . .. The relations are most conveniently written in terms of the generating series

T_{ij}(u) = δ_{ij} + ∑_{k=1}^{∞} T_{ij}^{(k)}u^{−k} ,

and have the form

(u − v)[T_{ij}(u), T_{kl}(v)] = T_{kj}(v)T_{il}(u) − T_{kj}(u)T_{il}(v) .

The coproduct for Y (𝔤𝔩_{2}) is given by

T_{ij}(u) ↦ ∑_{k=1}^{2} T_{kj}(u) ⊗ T_{ik}(u) .

The Yangian Y (𝔤𝔩_{2}) contains U(𝔤𝔩_{2}) as a Hopf subalgebra, the embedding U(𝔤𝔩_{2}) → Y (𝔤𝔩_{2}) being given by E_{ij} ↦ T_{ji}^{(1)}. We identify U(𝔤𝔩_{2}) wih its image in Y (𝔤𝔩_{2}) under this embedding.

There is a family of homomorphisms ϵ_{z} : Y (𝔤𝔩_{2}) → U(𝔤𝔩_{2}), depending on a complex parameter z:

ϵ_{z} : T_{ij}(u) ↦ δ_{ij} + \frac{E_{ji}}{u − z} .

These homomorphism are identical on the subalgebra U(𝔤𝔩_{2}) ⊂ Y (𝔤𝔩_{2}).

For any 𝔤𝔩_{2}-module V, we denote V (z) the pullback of V through the homomorphism ϵ_{z}; the Y (𝔤𝔩_{2})-module V (z) is called an evaluation module.

The Yangian action in a tensor product of evaluation modules is the underlying structure for the functional realization ϕ[l] of the tensor product of the corresponding 𝔤𝔩_{2}-modules.

As before, l = (l_{1}, l_{2}, l_{3}) will always denote a triple of nonnegative integers. For any k ∈ ℤ_{≥0} set

Z_{k} = {(k_{1}, k_{2}, k_{3}) ∈ ℤ_{≥0}^{3} | k_{1} + k_{2} + k_{3} = k} ,
Z_{k}[l] = {(k_{1}, k_{2}, k_{3}) ∈ Z_{k} | k_{1} ≤ l_{1}, k_{2} ≤ l_{2}, k_{3} ≤ l_{3}}.

Proposition 27. [KBI, TV3] Let z = (z_{1}, z_{2}, z_{3}) be a triple of complex numbers. Consider the Y (𝔤𝔩_{2})-module V_{(l_{1},0)}(z_{1}) ⊗ V_{(l_{2},0)}(z_{2}) ⊗ V_{(l_{3},0)}(z_{3}). For any k ∈ ℤ_{≥0}, q ∈ Z_{k}[l], we have

(∏_{i=1}^{3} ∏_{a=1}^{k}(t_{a} − z_{i})) T_{12}(t_{1}) … T_{12}(t_{k})(1_{l_{1}} ⊗ 1_{l_{2}} ⊗ 1_{l_{3}}) =

∑_{p∈Z_{k}[l]} w_{p}[l; z](t_{1}, …, t_{k})E_{12}^{p_{1}}1_{l_{1}} ⊗ E_{12}^{p_{2}}1_{l_{2}} ⊗ E_{12}^{p_{3}}1_{l_{3}},

(∏_{i=1}^{3} ∏_{a=1}^{k}(t_{a} − z_{i})) T_{21}(t_{1}) … T_{21}(t_{k})(E_{12}^{q_{1}}1_{l_{1}} ⊗ E_{12}^{q_{2}}1_{l_{2}} ⊗ E_{12}^{q_{3}}1_{l_{3}}) =

w′_{q}[l; z](t_{1}, …, t_{k})(1_{l_{1}} ⊗ 1_{l_{2}} ⊗ 1_{l_{3}}) , where

w_{k}[l; z](t_{1}, …, t_{k}) =

\frac{1}{k_{1}!k_{2}!k_{3}!} Sym (∏_{i=1}^{3} ( ∏_{a=1}^{k_{1}+···+k_{i−1}}(t_{a} − z_{i} + l_{i}) ∏_{b=k_{1}+···+k_{i}+1}^{k_{1}+k_{2}+k_{3}}(t_{b} − z_{i})) ∏_{a<b} \frac{t_{a} − t_{b} − 1}{t_{a} − t_{b}})

w′_{k}[l; z](t_{1}, …, t_{k}) =

\frac{1}{k_{1}!k_{2}!k_{3}!} Sym (∏_{i=1}^{3} ( ∏_{a=1}^{k_{1}+···+k_{i−1}}(t_{a} − z_{i}) ∏_{b=k_{1}+···+k_{i}+1}^{k_{1}+k_{2}+k_{3}}(t_{b} − z_{i} + l_{i})) ∏_{a<b} \frac{t_{a} − t_{b} + 1}{t_{a} − t_{b}}) .

The functions w_{k}[l; z] and w′_{k}[l; z] are called the weight functions and the dual weight functions, respectively. They are polynomials in t_{1}, …, t_{k}, and z_{1}, z_{2}, z_{3}.

The definition (6.1) of the the map ϕ[l] can be written as

ϕ[l; z] (E_{21}^{k_{1}}1_{l_{1}} ⊗ E_{21}^{k_{2}}1_{l_{2}} ⊗ E_{21}^{k_{3}}1_{l_{3}}) = \frac{k_{1}!l_{1}!}{(l_{1} − k_{1})!} \frac{k_{2}!l_{2}!}{(l_{2} − k_{2})!} \frac{k_{3}!l_{3}!}{(l_{3} − k_{3})!} w′_{k}[l; z] .

Let H′_{k} denote the space of symmetric polynomials in variables t_{1}, …, t_{k} of degree at most 2 in each t_{1}, …, t_{k}. Set

H′ = ⨁_{k∈ℤ_{≥0}} H′_{k} .

It is clear that H_{k} = H′_{k} ⊗ ℂ[z] and H = H′ ⊗ ℂ[z].

Theorem 28. [TV3] The space H′ has a structure of a Y (𝔤𝔩_{2})-module, depending on l, z, with the following properties:

1. the 𝔤𝔩_{2}-module structure on H′ coincides with that, induced by the 𝔤𝔩_{2}-module structure on H, described in Proposition 12;

2. the map ϕ[l; z] : V_{(l_{1},0)}(z_{1}) ⊗ V_{(l_{2},0)}(z_{2}) ⊗ V_{(l_{3},0)}(z_{3}) → H′ is a homomorphism of Y (𝔤𝔩_{2})- modules.

We now use the representation theory of the Yangian Y (𝔤𝔩_{2}) to prove the results, announced in Chapter 6. The following theorems go back to [T].

Theorem 29. [T]

1. Let l ∈ ℤ_{≥0}, and let U be an irreducible finite-dimensional Y (𝔤𝔩_{2})-module. Then for generic z ∈ ℂ, the Y (𝔤𝔩_{2})-modules U ⊗ V_{(l,0)}(z) and V_{(l,0)}(z) ⊗ U are irreducible and isomorphic to each other.

2. Let l, l′ ∈ ℤ_{≥0}, and let z, z′ ∈ ℂ be such that z − l = z′ − r for some r = 0, 1, …, min(l, l′) − 1. Then the Y (𝔤𝔩_{2})-module V_{(l,0)}(z) ⊗ V_{(l′,0)}(z′) contains a unique proper submodule Ṽ_{l,l′;r}(z). As a 𝔤𝔩_{2}-module, Ṽ_{l,l′;r}(z) is isomorphic to

Ṽ_{l,l′;s}(z) ≅ ⨁_{s=r+1}^{min(l,l′)} V_{(l+l′−s,s)}.  (B.1)

Theorem 30. [T] Let l, l′ ∈ ℤ_{≥0}. There exists a unique linear operator R_{l,l′}(z) ∈ End_{ℂ}(V_{(l,0)}⊗ V_{(l′,0)}), rationally depending on a complex parameter z, such that for generic z, z′ the operator

Ř_{l,l′}(z − z′) = P ∘ R_{l,l′}(z − z′) : V_{l}(z) ⊗ V_{l′}(z′) → V_{l′}(z′) ⊗ V_{l}(z)

is an isomorphism of Y (𝔤𝔩_{2})-modules, P being the permutation operator: P (u ⊗ v) = v ⊗ u, and

R_{l,l′}(z)1_{l} ⊗ 1_{l′} = 1_{l} ⊗ 1_{l′} .

We call the operator R_{l,l′}(z) the standard rational R-matrix.

Proof of Proposition 16. Let z_{0} = (z_{1}, z_{2}, z_{3}) be a generic point from the hyperplane z_{1}−z_{2}+ l_{2}−r = 0, with r = 0, 1, …, min(l_{1}, l_{2})−1. Then it follows from Theorem 29 that the Y (𝔤𝔩_{2})- module V_{(l_{1},0)}(z_{1})⊗V_{(l_{2},0)}(z_{2})⊗V_{(l_{3},0)}(z_{3}) has a unique proper submodule Ṽ_{l_{1},l_{2};r}(z_{1})⊗V_{(l_{3},0)}(z_{3}), while Proposition 27 and formula (B.1) imply that ker ϕ[l; z_{0}] contains Ṽ_{l_{1},l_{2};r}(z_{1})⊗V_{(l_{3},0)}(z_{3}). Since ϕ[l; z_{0}] is a non-trivial homomorphism of Y (𝔤𝔩_{2})-modules, the kernel ker ϕ[l; z_{0}] must be a proper Y (𝔤𝔩_{2})-submodule, which proves the equality

ker ϕ[l; z_{0}] = Ṽ_{l_{1},l_{2};r}(z_{1}) ⊗ V_{(l_{3},0)}(z_{3}) .

The proof of the second part of Proposition 16 is similar.

Proof of Proposition 18. It follows from Proposition 27 that to determine the image of a vector v ∈ V_{l_{1}}⊗V_{l_{2}}⊗V_{l_{3}}, we must act on it by the element (∏_{i=1}^{3} ∏_{a=1}^{k}(t_{a} − z_{i})) T_{21}(t_{1}) … T_{21}(t_{k}), and then use the linear functional (1_{l_{1}} ⊗ 1_{l_{2}} ⊗ 1_{l_{3}})^{∗}. The Y (𝔤𝔩_{2})-intertwining property of the R-matrix implies that we have the commutative diagram

V_{l_{2}} ⊗ V_{l_{1}} ⊗ V_{l_{3}} ⟶^{ϕ[s_{1}(l);s_{1}(z)]} H[s_{1}(l); s_{1}(z)] ↑^{R_{l_{1}l_{2}}(z_{1}−z_{2})⊗1}  ∥ V_{l_{1}} ⊗ V_{l_{2}} ⊗ V_{l_{3}}  ⟶^{ϕ[l;z]}  H[l; z] ↓^{1⊗R_{l_{2}l_{3}}(z_{2}−z_{3})}  ∥ V_{l_{1}} ⊗ V_{l_{3}} ⊗ V_{l_{2}} ⟶^{ϕ[s_{2}(l);s_{2}(z)]} H[s_{2}(l); s_{2}(z)] .  (B.2)

Now denote

Ř_{l,l′}(u) = \frac{∏_{j=1}^{l′}(u − l + j)}{∏_{j=1}^{l}(−u − l′ + j)} Ř_{l,l′}(u) .

The commutative diagram (B.2) and the formulas

\frac{∏_{j=1}^{l_{2}}(z_{1} − z_{2} − l_{1} + j)}{∏_{j=1}^{l_{1}}(z_{2} − z_{1} − l_{2} + j)} = \frac{X [l; z]}{X [s_{1}(l); s_{1}(z)]} ,

\frac{∏_{j=1}^{l_{3}}(z_{2} − z_{3} − l_{2} + j)}{∏_{j=1}^{l_{2}}(z_{3} − z_{2} − l_{3} + j)} = \frac{X [l; z]}{X [s_{2}(l); s_{2}(z)]} ,

imply that the operators Ř_{l,l′}(u) satisfy the conditions of Proposition 18.

In the remaining part of the Appendix we construct the map I[l] explicitly, and describe its singularities.

Let u = (u_{1}, u_{2}, u_{3}) be complex variables, and for any k ∈ ℤ_{≥0} set H̃_{k} = H_{k} ⊗ ℂ[u]. Denote also

Ω_{k}[u; z](t_{1}, …, t_{k}) = ∏_{i=1}^{3} ∏_{a=1}^{k} \frac{1}{(t_{a} − z_{i} + u_{i})(t_{a} − z_{i})} ∏_{\begin{matrix}a,b=1 \\ a≠b \end{matrix}}^{k} \frac{t_{a} − t_{b}}{t_{a} − t_{b} − 1} .

For any function F = F (t_{1}, …, t_{k}), set

Res_{(ζ_{1},...,ζ_{k})}F = Res_{t_{k}=ζ_{k}} (Res_{t_{k−1}=ζ_{k−1}} (… Res_{t_{1}=ζ_{1}}F (t_{1}, …, t_{k})) …) ,

Introduce a symmetric ℂ(u; z)-valued bilinear form ⟨·, ·⟩ on each H̃_{k}, so that for any φ, ψ ∈ H̃_{k} the value ⟨φ, ψ⟩_{u,z} of the corresponding rational function at generic u, z is given by

⟨φ, ψ⟩_{u,z} = ∑_{k∈Z_{k}} Res_{τ_{k}^{−}[z]} (φ[u; z]ψ[u; z]Ω_{k}[u; z]) ,  (B.3)

where the “string” τ_{k}^{−}[z] is defined by

τ_{k}^{−}[z] = (z_{1}, …, z_{1} − k_{1} + 1, z_{2}, …, z_{2} − k_{2} + 1, z_{3}, …, z_{3} − k_{3} + 1) .

Remark. The definition above is motivated by the fact (see [TV4]) that the integral over the imaginary hyperplane Re t_{1} = ⋯ = Re t_{k} = 0,

\frac{1}{(2πi)^{k}} ∫ ⋯ ∫ φ[u; z](t_{1}, …, t_{k})ψ[u; z](t_{1}, …, t_{k})Ω_{k}[u; z](t_{1}, …, t_{k})dt_{1} … dt_{k} ,

converges when Re(u_{i}) ≪ Re(z_{i}) < 0, i = 1, 2, 3, and represents the rational function ⟨φ, ψ⟩.

Evaluating the integral by residues, and using the analytic continuation, we get the more convenient algebraic formula (B.3); one can show that it is equivalent to another iterated residue formula:

⟨φ, ψ⟩_{u,z} = (−1)^{k} ∑_{k∈Z_{k}} Res_{τ_{k}^{+}[z−u]} (φ[u; z]ψ[u; z]Ω_{k}[u; z]) ,  (B.4)

where τ_{k}^{+}[z − u] is defined by

τ_{k}^{+}[z−u] = (z_{1}−u_{1}, …, z_{1}−u_{1}+k_{1}−1, z_{2}−u_{2}, …, z_{2}−u_{2}+k_{2}−1, z_{3}−u_{3}, …, z_{3}−u_{3}+k_{3}−1).

Lemma 31. For any φ, ψ ∈ H̃_{k}, the only possible singularities of ⟨φ, ψ⟩_{u,z} are simple poles at the following hyperplanes:

z_{i} − u_{i} = z_{j} − r,  i, j = 1, 2, 3,  r = 0, 1, …, k − 1.  (B.5)

Proof. Since φ, ψ are polynomials, possible singularities of ⟨φ, ψ⟩_{u,z} are determined by Ω_{k}[u; z]. Using explicit formulas, one can check that Res_{τ_{k}^{−}[z]}(Ω_{k}[u; z]) has only simple poles located at the hyperplanes

z_{i} − u_{i} = z_{j} − r,  z_{i} = z_{j} − r,  i, j = 1, 2, 3,  r = 0, 1, …, k − 1 .

Similarly, Res_{τ_{k}^{+}[z−u]}(Ω_{k}[u; z]) has only simple poles located at the hyperplanes

z_{i} − u_{i} = z_{j} − r,  z_{i} = z_{j} − u_{j} − r,  i, j = 1, 2, 3,  r = 0, 1, …, k − 1 .

Since ⟨φ, ψ⟩_{u,z} is given either by formula (B.3) or (B.4), the function ⟨φ, ψ⟩_{u,z} can have poles only at the hyperplanes which belong to both of these lists, which gives precisely list (B.5).

Let H̃_{k}[l], k ∈ ℤ_{≥0}, denote the subspace of functions φ ∈ H̃_{k}, satisfying the following conditions for each i = 1, 2, 3 such that l_{i} < k:

φ(z_{i}, …, z_{i} − l_{i}, t_{l_{i}+2}, …, t_{k}) = 0 at the hyperplane u_{i} = l_{i} .

We have two important imbeddings of H_{k}[l] into H̃_{k}[l], given by φ ↦ φ^{−},

φ^{−}[u; z](t_{1}, …, t_{k}) = φ[z](t_{1}, …, t_{k}),

and φ ↦ φ^{+},

φ^{−}[u; z](t_{1}, …, t_{k}) = φ[z − u + l](t_{1}, …, t_{k}).

For p ∈ Z_{k}, let w_{p} = w_{p}[u; z] be the weight function defined in Proposition 27, with integers l = (l_{1}, l_{2}, l_{3}) replaced by variables u = (u_{1}, u_{2}, u_{3}). It is obvious that w_{p} ∈ H̃_{k}; moreover, if p ∈ Z_{k}[l], then w_{p} ∈ H̃_{k}[l].

Lemma 32. Let p ∈ Z_{k}[l], and φ ∈ H_{k}[l].

1. The function ⟨w_{p}, φ^{−}⟩_{u,z} may have only simple poles, located at the hyperplanes

z_{i} − u_{i} = z_{j} − r,  1 ≤ i ≤ j ≤ 3,  r = 0, 1, …, l_{j} − 1.

2. The function ⟨w_{p}, φ^{+}⟩_{u,z} may have only simple poles, located at the hyperplanes

z_{i} − u_{i} = z_{j} − r,  1 ≤ i ≤ j ≤ 3,  r = 0, 1, …, l_{i} − 1.

Proof. Since φ^{−}[u; z](τ_{q}^{−}[z]) = φ[z](τ_{q}^{−}[z]) = 0 unless q ∈ Z_{k}[l], we have

⟨w_{p}, φ^{−}⟩_{u,z} = ∑_{q∈Z_{k}} φ^{−}(τ_{q}^{−}[z])Res_{τ_{q}^{−}[z]}(w_{p}[u; z]Ω_{k}[u; z]) =
∑_{q∈Z_{k}[l]} φ^{−}(τ_{q}^{−}[z])Res_{τ_{q}^{−}[z]}(w_{p}[u; z]Ω_{k}[u; z]) .

Let q ∈ Z_{k}[l]. From the explicit formulas for w_{p}[u; z] and Ω_{k}[u; z] it follows that the function Res_{τ_{q}^{−}[z]}(w_{p}[u; z]Ω_{k}[u; z]) has no pole at the hyperplane z_{i} − u_{i} = z_{j} − r provided i > j or r ≥ q_{j}. Using the description (B.5) of all possible poles, we see that ⟨w_{p}, φ^{−}⟩_{u,z} may have only poles corresponding to i ≤ j and r < q_{j} ≤ l_{j}, which proves the first assertion.

Similarly, using the fact that φ^{+}[u; z](τ_{q}^{+}[z])φ[z − u](τ_{q}^{+}[z − u]) = 0 unless q ∈ Z_{k}[l], we arrive to the formula

⟨w_{p}, φ^{+}⟩_{u,z} = ∑_{q∈Z_{k}[l]} φ^{+}(τ_{q}^{+}[z − u])Res_{τ_{q}^{+}[z−u]}(w_{p}[u; z]Ω_{k}[u; z]) .

For any q ∈ Z_{k}[l] the function Res_{τ_{q}^{+}[z−u]}(w_{p}[u; z]Ω_{k}[u; z]) has no pole at the hyperplane z_{i} − u_{i} = z_{j} − r provided i > j or r ≥ q_{i}. Therefore, it follows that ⟨w_{p}, φ^{+}⟩_{u;z} may only have poles corresponding to i ≤ j and r < q_{i} ≤ l_{i}.

Lemma 33. Let φ, ψ ∈ H̃_{k}[l]. The function ⟨φ, ψ⟩_{u,z} analytically continues to u = l, and we have

⟨φ, ψ⟩_{l,z} = ∑_{k∈Z_{k}[l]} Res_{τ_{k}^{−}[z]} (φ[l; z]ψ[l; z]Ω_{k}[l; z])
= (−1)^{k} ∑_{k∈Z_{k}[l]} Res_{τ_{k}^{+}[z−l]} (φ[l; z]ψ[l; z]Ω_{k}[l; z]) .  (B.6)

In other words, the analytic continuation to u = l can be computed by applying the residue formula to the specialized functions ψ[l; z], φ[l; z] and Ω_{k}[l; z], rather than treating u as variables, and the summation is reduced to k ∈ Z_{k}[l] ⊂ Z_{k}. Note that in general this is not true for arbitrary φ, ψ ∈ H̃_{k}.

Proof. One can check that for any q ∈ Z_{k} the rational functions Res_{τ_{q}^{−}[z]}(ψ[u; z]Ω_{k}[u; z]) and Res_{τ_{q}^{+}[z−u]}(ψ[u; z]Ω_{k}[u; z]) have no poles at the hyperplanes u_{i} = l_{i}, i = 1, 2, 3. Together with Lemma 31, this shows that the analytic continuation to u = l is well-defined.

Since for q ∉ Z_{k}[l], the expression φ[u; z](τ_{k}^{−}[z]) vanishes at u = l, it follows that the summation in the residue formulas (B.3) reduces to q ∈ Z_{k}[l]. Similar conclusion holds for the redisue formula (B.4).

The statement now follows from the fact that for q ∈ Z_{k}[l] we have

Res_{τ_{q}^{−}[z]}(ψ[u; z]Ω_{k}[u; z])|_{u=l} = Res_{τ_{q}^{−}[z]}(ψ[l; z]Ω_{k}[l; z]) ,
Res_{τ_{q}^{+}[z−u]}(ψ[u; z]Ω_{k}[u; z])|_{u=l} = Res_{τ_{q}^{+}[z−l]}(ψ[l; z]Ω_{k}[l; z]) .

Introduce a partial order ≤ on Z_{k} by saying that x ≤ y if and only if x_{1} ≤ y_{1} and x_{3} ≥ y_{3}.

Lemma 34. For any k, p, q ∈ Z_{k} we have

w_{p}[l, z](τ_{k}^{−}[z]) = 0, unless k ≤ p ,
w′_{q}[l, z](τ_{k}^{−}[z]) = 0, unless k ≥ q .

Moreover, if z is generic, then w′_{q}[l, z](τ_{q}^{−}[z]) ≠ 0.

The proof is straightforward.

Lemma 35. [TV4] Let k ∈ ℤ_{≥}0, and let p, q ∈ Z_{k}[l]. Then, we have

⟨w_{p}, w′_{q}⟩_{l,z} = δ_{p,q} \frac{p_{1}!l_{1}!}{(l_{1} − p_{1})!} \frac{p_{2}!l_{2}!}{(l_{2} − p_{2})!} \frac{p_{3}!l_{3}!}{(l_{3} − p_{3})!} .  (B.7)

Proof. By Lemma 34 the summation in the residue formula (B.6) reduces to k such that q ≤ k ≤ p, that is

⟨w_{p}, w′_{q}⟩_{l,z} = ∑_{q≤k≤p} w′_{q}[l; z](τ_{k}^{−}[z])w_{p}[l; z](τ_{k}^{−}[z])Res_{τ_{k}^{−}[z]}(Ω_{k}[l; z]) .

It follows immediately that ⟨w_{p}, w′_{q}⟩_{l,z} = 0 unless q ≤ p.

Similarly, one can check that

w_{p}(τ_{k}^{+}[z − l]) = 0, unless k ≥ p ,
w′_{q}(τ_{k}^{+}[z − l]) = 0, unless k ≤ q ,

Thus, it follows that

⟨w_{p}, w′_{q}⟩_{l,z} = ∑_{p≤k≤q} w_{p}[l; z](τ_{k}^{+}[z − l])w′_{q}[l; z](τ_{k}^{+}[z − l]) (Res_{τ_{k}^{+}[z−l]}Ω_{k}[l, z]) .

and therefore, ⟨w_{p}, w′_{q}⟩_{l,z} = 0 unless p ≤ q.

Hence, ⟨w_{p}[l, z], w′_{q}[l, z]⟩_{l,z} is equal to zero unless p = q. In the last case there is only one term in the residue summation formula, corresponding to k = p = q, and the statement follows from the direct computation of the only remaining residue at τ_{k}[z].

Finally, we have all ingredients necessary to describe explicitly the map I[l] and to prove its properties.

Proof of Proposition 14. Lemma 34 implies that the dual weight functions {w′_{k}[l; z]} are linearly independent as elements of H_{k}, which yields the injectivity of the map ϕ[l]. Using the explicit combinatorial formulas for w′_{k}[l; z], one can check that the admissibility conditions (6.2) are satisfied provided that k ∈ Z_{k}[l]. Hence, Im ϕ[l] ⊗_{ℂ[z]} ℂ(z) ⊂ H_{k}[l] ⊗_{ℂ[z]} ℂ(z). To show that these spaces coincide, it is enough to check that for generic z one has dim H_{k}[l; z] ≤ dim Im ϕ[l; z] = #Z_{k}[l].

Consider the maps ε_{p}[z] : H′_{k} → ℂ,

ε_{p}[z] : ψ ↦ ψ(τ_{p}[z]) .

Lemma 34 implies that for generic z these maps are linearly independent. It is clear that for any ψ ∈ H[l; z], p ∉ Z_{k}[l] one has ε_{p}[z](ψ) = 0, which gives dim H_{k}[l; z] ≤ #Z_{k}[l], since dim H′_{k} = #Z_{k}.

Now, consider the map

I[l; z] : H[l; z] → V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)} ,

I[l; z]φ = ∑_{k∈ℤ_{≥_{0}}} ∑_{p∈Z_{k}[l]}⟨w_{p}, φ⟩_{l,z}E_{21}^{p_{1}}1_{l_{1}} ⊗ E_{21}^{p_{2}}1_{l_{2}} ⊗ E_{21}^{p_{3}}1_{l_{3}} .

Formulas (6.1) and (B.7) imply (6.4).

According to Lemma 33, we have

⟨w_{p}, φ⟩_{l,z} = (⟨w_{p}, φ^{−}⟩_{u,z}) |_{u=l} (⟨w_{p}, φ^{+}⟩_{u,z}) |_{u=l} ,

and Lemma 32 implies that ⟨w_{p}, φ⟩_{l,z} may have poles only at the hyperplanes

z_{i} − u_{i} = z_{j} − r, 1 ≤ i < j ≤ 3, r = 0, 1, …, min(l_{i}, l_{j}) − 1 .

This proves the last part of Proposition 14.

Appendix C.

In this section we prove the existence of the distinguished polynomial basis {ς_{i}[l]} of H_{k}[l]^{sing}, see Lemma 19.

Proof of Lemma 19. The subspace H_{k}[l]^{sing} consists of polynomials of degree at most 1 in each t_{j}, and therefore any function ς(t_{1}, …, t_{k}) ∈ H_{k}[l]^{sing} has the form

ς(t_{1}, …, t_{k}) = ∑_{r=0}^{k} X_{r}(z_{1}, z_{2}, z_{3}) σ_{r}(t_{1}, …, t_{k})

where X_{0}, … X_{k} are some coefficients from ℂ[z], and

σ_{r}(t_{1}, …, t_{k}) = ∑_{1≤i_{1}<···<i_{r}≤k} t_{i_{1}} … t_{i_{r}}, r = 0, 1, 2, …;

in particular, σ_{0}(t_{1}, …, t_{k}) ≡ 1, and σ_{r}(t_{1}, …, t_{k}) ≡ 0 if r > k.

Using the combinatorial identity

σ_{r}(z, …, z − l, t_{l+2}, …, t_{k}) = ∑_{s=0}^{r} σ_{r−s}(z, …, z − l)σ_{s}(t_{l+2}, …, t_{k}) ,

we see that admissibility conditions (6.2) for the function ς(t_{1}, …, t_{k}) are equivalent to the following system of equations for the coefficients X_{0}, … X_{k}:

∑_{r=s}^{k} X_{r} σ_{r−s}(z_{i}, …, z_{i} − l_{i}) = 0,  s = 0, …, k − l_{i} − 1, i = 1, 2, 3.  (C.1)

Let d = dim (V_{(l_{1},0)} ⊗ V_{(l_{2},0)} ⊗ V_{(l_{3},0)})^{sing} [(l_{1} + l_{2} + l_{3} − k, k)]. We will show that system (C.1) determines X_{0}, …, X_{k−d} as linear combinations of free variables X_{k−d+1}, …, X_{k} with coefficients, polynomial in z_{1}, z_{2}, z_{3}. Then we construct ς_{s}[l], s = 1, …, d, by taking

X_{k−s+1} = 1, X_{k−r+1} = 0 for r ≠ s, 1 ≤ r ≤ d ,

and computing X_{0}, …, X_{k−d} from equations (C.1). The polynomials ς_{s}[l] obeys invariance (7.2) by construction.

Consider the (k − d + 1) × (k + 1) matrix of system (C.1). It suffices to prove that its (k − d + 1) × (k − d + 1) minor, corresponding to the variables X_{0}, …, X_{k−d}, is nonzero, and divides all other (k − d + 1) × (k − d + 1) minors. Before giving a general proof of this statement, we consider an example.

Example: l_{1} = 1, l_{2} = 2, k = 3, l_{3} > 3. Then d = dim (V_{(1,0)} ⊗ V_{(2,0)} ⊗ V_{(l_{3},0)})^{sing} [(l_{3}, 3)] = 1, and the matrix of system of (C.1) is equal to

(\begin{matrix}1 & z_{1} + (z_{1} − 1) & z_{1}(z_{1} − 1) & 0 \\ 0 & 1 & z_{1} + (z_{1} − 1) & z_{1}(z_{1} − 1) \\ 1 & z_{2} + (z_{2} − 1) + (z_{2} − 2) & z_{2}(z_{2} − 1) + z_{2}(z_{2} − 2) + (z_{2} − 1)(z_{2} − 2) & z_{2}(z_{2} − 1)(z_{2} − 2) \end{matrix}) .

The 3 × 3 minors of this matrix are given by

M_{012} = 3(z_{1} − z_{2})(z_{1} − z_{2} + 1) ,
M_{013} = (z_{1} − z_{2})(z_{1} − z_{2} + 1)(2z_{1} + z_{2} − 2) ,
M_{023} = (z_{1} − z_{2})(z_{1} − z_{2} + 1)(z_{1}^{2} + 2z_{1}z_{2} − 3z_{1} − z_{2} + 2) ,
M_{123} = (z_{1} − z_{2})(z_{1} − z_{2} + 1)(3z_{1}^{2}z_{2} − 3z_{1}^{2} − 3z_{1}z_{2} + 5z_{1} + z_{2} − 2) .

The leftmost minor M_{012} is nonzero and divides all other minors. The function ς(t_{1}, t_{2}, t_{3}), spanning the space H[(1, 2, l_{3}), (z_{1}, z_{2}, z_{3})]^{sing}[(l_{3}, 3)], is given by

ς(t_{1}, t_{2}, t_{3}) = t_{1}t_{2}t_{3} − \frac{1}{M_{012}} (M_{013}(t_{1}t_{2} + t_{2}t_{3} + t_{1}t_{3}) − M_{023}(t_{1} + t_{2} + t_{3}) + M_{123}) .

We now return to the case of arbitrary l. Without loss of generality, we may assume that l_{1} ≤ l_{2} ≤ l_{3}. To shorten notation, we denote

σ_{r}^{(i)} = σ_{r}(z_{i}, z_{i} − 1, …, z_{i} − l_{i}), r = 0, …, k, i = 1, 2, 3.

Case 1: k ≤ l_{1}. Then, there are no admissibility conditions, d = k + 1, and all the variables X_{0}, … X_{k} are free.

Case 2: l_{1} < k ≤ l_{2}. Then d = l_{1} + 1, and the matrix of system (C.1) is equal to

(\begin{matrix}σ_{0}^{(1)} & σ_{1}^{(1)} & σ_{2}^{(1)} & … & σ_{k−l_{1}−1}^{(1)} & σ_{k−l_{1}}^{(1)} & σ_{k−l_{1}+1}^{(1)} & … & σ_{k}^{(1)} \\ 0 & σ_{0}^{(1)} & σ_{1}^{(1)} & … & σ_{k−l_{1}−2}^{(1)} & σ_{k−l_{1}−1}^{(1)} & σ_{k−l_{1}}^{(1)} & … & σ_{k−1}^{(1)} \\ 0 & 0 & σ_{0}^{(1)} & … & σ_{k−l_{1}−3}^{(1)} & σ_{k−l_{1}−2}^{(1)} & σ_{k−l_{1}−1}^{(1)} & … & σ_{k−2}^{(1)} \\ ⋮ & ⋮ & ⋮ & ... & … & … & … & ... & … \\ 0 & 0 & 0 & … & σ_{0}^{(1)} & σ_{1}^{(1)} & σ_{2}^{(1)} & … & σ_{l_{1}+1}^{(1)} \end{matrix})

Note that some of the terms in the upper right corner can also vanish, because σ_{l_{1}+2}^{(1)} = ⋯ = σ_{k}^{(1)} = 0.

Since σ_{0}^{(1)} ≡ 1, the leftmost (k − d + 1) × (k − d + 1) minor is identically equal to 1. Therefore X_{k−d+1}, …, X_{k} are free variables, and X_{0}, …, X_{k−d} are their linear combinations with coefficients, polynomial in z_{1}, z_{2}, z_{3}.

Case 3: l_{2} < k ≤ l_{3}. If k ≥ l_{1} + l_{2}, then d = 0, and there is nothing to check. Assume that k < l_{1} + l_{2}, which gives d = l_{1} + l_{2} − k. The matrix of system (C.1) is equal to

(\begin{matrix}σ_{0}^{(1)} & σ_{1}^{(1)} & … & … & … & σ_{k−l_{1}−1}^{(1)} & … & σ_{k}^{(1)} \\ 0 & σ_{0}^{(1)} & … & … & … & σ_{k−l_{1}−2}^{(1)} & … & σ_{k−1}^{(1)} \\ ⋮ & ⋮ & ⋮ & ... & … & … & ... & … \\ 0 & 0 & … & … & … & σ_{0}^{(1)} & … & σ_{l_{1}+1}^{(1)} \\ σ_{0}^{(2)} & σ_{1}^{(2)} & … & σ_{k−l_{2}−1}^{(2)} & … & … & … & σ_{k}^{(2)} \\ 0 & σ_{0}^{(2)} & … & σ_{k−l_{2}−2}^{(2)} & … & … & … & σ_{k−1}^{(2)} \\ ⋮ & ⋮ & ... & … & … & … & ... & … \\ 0 & 0 & … & σ_{0}^{(2)} & … & … & … & σ_{l_{2}+1}^{(2)} \end{matrix})  (C.2)

We make two claims now. First, every (k − d + 1) × (k − d + 1) minor of this matrix is a polynomial in z_{1}, z_{2}, divisible by

D(z_{1}, z_{2}) = ∏_{j_{1}=1}^{k−l_{1}} ∏_{j_{2}=1}^{k−l_{2}}(z_{1} − z_{2} + j_{1} − j_{2}) .

Second, the leftmost (k − d + 1) × (k − d + 1) minor of matrix (C.2) is a nonzero polynomial in z_{1}, z_{2} of homogeneous degree (k − l_{1})(k − l_{2}).

These two claims together imply the desired statement. Indeed, since the homogeneous degree of the polynomial D(z_{1}, z_{2}) is equal to (k−l_{1})(k−l_{2}), the leftmost minor is proportional to D(z_{1}, z_{2}) with a nonzero coefficient. Therefore, every (k − d + 1) × (k − d + 1) minor is divisible by the leftmost (k − d + 1) × (k − d + 1) minor, which means that X_{0}, …, X_{k−d} can be expressed as linear combinations of the free variables X_{k−d+1}, …, X_{k} with coefficients, polynomially depending on z_{1}, z_{2}, z_{3}.

We first prove the second claim. Note that σ_{r}^{(i)} is a polynomial of z_{i} with the highest degree term \binom{l_{i}+1}{r}z_{i}^{r}. Therefore, the homogeneous degree of the leftmost (k − d + 1) × (k − d + 1) minor does not exceed (k − l_{1})(k − l_{2}), while the sum of terms of degree (k − l_{1})(k − l_{2}) is given by the determinant

det (\begin{matrix}1 & \binom{l_{1}+1}{1}z_{1} & … & … & … & \binom{l_{1}+1}{k−l_{1}−1}z_{1}^{k−l_{1}−1} & … & \binom{l_{1}+1}{k−d−1}z_{1}^{k−d−1} \\ 0 & 1 & … & … & … & \binom{l_{1}+1}{k−l_{1}−2}z_{1}^{k−l_{1}−2} & … & \binom{l_{1}+1}{k−d−2}z_{1}^{k−d−2} \\ ⋮ & ⋮ & ⋮ & ... & … & … & ... & … \\ 0 & 0 & … & … & … & 1 & … & \binom{l_{1}+1}{k−l_{2}}z_{1}^{k−l_{2}} \\ 1 & \binom{l_{2}+1}{1}z_{2} & … & \binom{l_{2}+1}{k−l_{2}−1}z_{2}^{k−l_{2}−1} & … & … & … & \binom{l_{2}+1}{k−d−1}z_{2}^{k−d−1} \\ 0 & 1 & … & \binom{l_{2}+1}{k−l_{2}−2}z_{2}^{k−l_{2}−2} & … & … & … & \binom{l_{2}+1}{k−d−2}z_{2}^{k−d−2} \\ ⋮ & ⋮ & ... & … & … & … & ... & … \\ 0 & 0 & … & 1 & … & … & … & \binom{l_{2}+1}{k−l_{1}}z_{2}^{k−l_{1}} \end{matrix})

The monomial z_{1}^{(k−l_{1})(k−l_{2})} enters this determinant with a nonzero coefficient

det (\begin{matrix}\binom{l_{1}+1}{k−l_{2}} & \binom{l_{1}+1}{k−l_{2}+1} & … & \binom{l_{1}+1}{2k−l_{1}−l_{2}−1} \\ \binom{l_{1}+1}{k−l_{2}−1} & \binom{l_{1}+1}{k−l_{2}} & … & \binom{l_{1}+1}{k−1} \\ ⋮ & ⋮ & ... & ⋮ \\ \binom{l_{1}+1}{l_{1}−l_{2}+1} & \binom{l_{1}+1}{l_{1}−l_{2}+2} & … & \binom{l_{1}+1}{k−l_{2}} \end{matrix}) = ∏_{i=1}^{k−l_{1}} ∏_{j=1}^{k−l_{2}} ∏_{s=1}^{l_{1}+l_{2}−k+1} \frac{i + j + s − 1}{i + j + s − 2} .

The last equality follows from the general formula, established in [K]:

det (\begin{matrix}\binom{a+b}{a} & \binom{a+b}{a+1} & … & \binom{a+b}{a+c} \\ \binom{a+b}{a−1} & \binom{a+b}{a} & … & \binom{a+b}{a+c−1} \\ ⋮ & ⋮ & ... & ⋮ \\ \binom{a+b}{a−c} & \binom{a+b}{a−c+1} & … & \binom{a+b}{a} \end{matrix}) = ∏_{i=1}^{c+1} ∏_{j=1}^{a} ∏_{s=1}^{b} \frac{i + j + s − 1}{i + j + s − 2} ,

by taking a = k − l_{2}, b = l_{1} + l_{2} − k + 1, c = k − l_{1} − 1.

Finally, we prove the first claim. Recall that we assume l_{1} ≤ l_{2} < k < l_{1} + l_{2}.

Take an integer j such that −l_{2} ≤ j ≤ l_{1} and let z_{1} − z_{2} − j = 0. Then the sets {z_{1}, z_{1} − 1, …, z_{1}−l_{1}} and {z_{2}, z_{2}−1, …, z_{2}−l_{2}} have p(j)+1 common points {ζ, ζ −1, …, ζ −p(j)}, where ζ = z_{1} for j < 0, ζ = z_{2} for j ≥ 0, and

p(j) = \{ \begin{matrix}l_{2} + j, & −l_{2} ≤ j < l_{1} − l_{2} , \\ l_{1}, & l_{1} − l_{2} ≤ j < 0 , \\ l_{1} − j, & 0 ≤ j ≤ l_{1} , \end{matrix}

Consider the system of linear equations on variables X_{0}, …, X_{k}, corresponding to the condition

ς(ζ, ζ − 1, …, ζ − p(j), t_{p(j)+2}, …, t_{k}) ≡ 0 .  (C.3)

Let L ⊂ ℂ^{k} denote the linear span of the rows of the corresponding matrix. Since the rows are linearly independent, we have dim L = k − p(j).

Obviously, if the function ς(t_{1}, …, t_{k}) obeys condition (C.3), then

ς(z_{1}, z_{1} − 1, …, z_{1} − l_{1}, t_{l_{1}+1}, …, t_{k}) ≡ 0 ,
ς(z_{2}, z_{2} − 1, …, z_{2} − l_{2}, t_{l_{2}+1}, …, t_{k}) ≡ 0 .

This means that the linear span of the rows of matrix (C.2) is contained in L. In particular, the rank of matrix (C.2) does not exceed dim L. Therefore, if l_{1} − k < j < k − l_{2}, then the rank of matrix (C.2) is less than its number of rows 2k − l_{1} − l_{2} by at least k + p(j) − l_{1} − l_{2}, which is positive.

The above consideration shows that every (k − d + 1) × (k − d + 1) minor of matrix (C.2) is divisible by the product

∏_{j=l_{1}−k+1}^{k−l_{2}−1}(z_{1} − z_{2} − j)^{k+p(j)−l_{1}−l_{2}} .

It is easy to show that this product coincides with D(z_{1}, z_{2}).

Case 4: l_{1} + l_{2} < k ≤ l_{3}. In this case dim H_{k}(l; z)^{sing} = 0, and there is nothing to check. Case 5: k > l_{3}. There are three types of admissibility conditions in this case, related

to (z_{1}, l_{1}), (z_{2}, l_{2}) and (z_{3}, l_{3}). The consideration, similar to Case 3, shows that the leftmost (k − d + 1) × (k − d + 1) minor of the corresponding matrix is nonzero and divides every (k − d + 1) × (k − d + 1) minor. This implies the required statement.

References

[BGG] J. Bernstein, I. Gelfand, S. Gelfand, A category of representations of Lie algebras, Func. Anal. Appl. 5 (1971), 1–8.

[CP] V. Chari, A. Pressley, A guide to quantum groups, Cambridge University Press, 1994.

[ES] P. Etingof, K. Styrkas, Algebraic integrability and representations of Lie algebras, Compositio Mathematica 98 (1995), 91–112.

[EV] P. Etingof, A. Varchenko, Dynamical Weyl groups and applications, QA/0011001.

[FFM] F. Malikov, B. Feigin, D. Fuks, Singular vectors in Verma modules over Kac-Moody algebras, Func. Anal. and Appl., Vol. 20, (1986).

[H] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, Israel Math. Conf. Proc. 8 (1995) 1–182.

[K] C. Krattenthaler, Advanced Determinant Calculus, Séminaire Lotharingien Combin. 42 (1999) (The Andrews Festschrift), paper B42, 67 pp.

[KBI] V. Korepin, N. Bogolyubov, A. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, 1993.

[T] V. Tarasov, Irreducible monodromy matrices for the R-matrix of the XXZ-model and lattice local quantum Hamiltonians, Theor. Math. Phys. 63 (1985) 440–454.

[TV] V. Tarasov, A. Varchenko, Difference equations compatible with trigonometric KZ equations, IMRN (2000), no. 15, 801–829.

[TV2] V. Tarasov, A. Varchenko, Duality for Knizhnik-Zamolodchikov and Dynamical Equations, Acta Appl. Math. (2002).

[TV3] V. Tarasov, A. Varchenko, Geometry of q-hypergeometric functions as a bridge between Yangians and quantum affine algebras, Inventiones Mathematicae 128 (1997) no. 3, 501–588.

[TV4] V. Tarasov, A. Varchenko, Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997) 1–135.

[Zh] D. Zhelobenko, Compact Lie groups and their representations, Translations of Mathematical Monographs 40, American Mathematical Society, 1973.