Cherednik algebras and Yangians

Nicolas Guay

Abstract

We construct a functor from the category of modules over the trigonometric (resp. rational) Cherednik algebra of type 𝔤𝔩_{l} to the category of integrable modules of level l over a Yangian for the loop algebra 𝔰𝔩_{n} (resp. over a subalgebra of this Yangian called the Yangian deformed double loop algebra) and we establish that it is an equivalence of categories if l + 2 < n.

1 Introduction

One of the most important classical results in representation theory is an equivalence, called Schur-Weyl duality, between the category of modules over the symmetric group S_{l} and the category of modules of level l over the Lie algebra 𝔰𝔩_{n} if l ≤ n − 1. When quantum groups were invented in the 1980’s, it became an interesting problem to generalize the Schur-Weyl correspondence and similar equivalences were obtained between finite Hecke algebras and quantized enveloping algebras [Ji], between degenerate affine Hecke algebras and Yangians [Dr1, Ch1], between affine Hecke algebras and quantized affine Lie algebras [ChPr1], and between double affine Hecke algebras and toroidal quantum algebras [VaVa]. In this paper, we prove a similar equivalence of categories between the trigonometric (resp. rational) Cherednik algebra associated to the symmetric group S_{l} and a (resp. subalgebra 𝕃 of a) Yangian LY for the loop algebra L𝔰𝔩_{n} = 𝔰𝔩_{n} ⊗_{ℂ} ℂ[u, u^{−1}].

The algebra on the other side of our equivalence from the Cherednik algebra is barely known. The author is not aware of any paper devoted to a study of Yangians of affine type; the only mention he could find of this notion anywhere in the literature is in [Va]. By contrast, there has been a recent surge of interest in the representation theory of Cherednik algebras and their relations to the geometry of Hilbert schemes, integrable systems and other important mathematical objects. Our duality theorem indicates a new route to those questions via a careful study of the Yangian and makes the study of this algebra more relevant and interesting.

Affine Hecke algebras are very important in representation theory and have been studied extensively over the last few decades, along with their degenerate (graded) version introduced in [Dr1] and in [Lu]. About fifteen years ago, I. Cherednik introduced the notion of double affine Hecke algebra [Ch2], abbreviated DAHA, which he used to prove important conjectures of I. Macdonald. His algebra also admits degenerate versions, the trigonometric one and the rational one, which are called Cherednik algebras. The trigonometric DAHA is generated by two subalgebras, one isomorphic to a degenerate affine Hecke algebra and the other one isomorphic to the group algebra of an affine Weyl group. For this reason, and because of the results mentioned above, we can expect its Schur-Weyl dual to be built from one copy of the Yangian Y for 𝔰𝔩_{n} and from one copy of the loop algebra L𝔰𝔩_{n}. This is indeed true for LY.

An epimorphic image of 𝕃, defined in terms of operators acting on a certain space, appeared for the first time in [BHW]; this was known to P. Etingof and V. Ginzburg. However, the algebra considered in that paper is not described in a very precise way and no equivalence of categories is established. One motivation for the present article comes from our desire to find exactly the relations between the generators of the Schur-Weyl dual of a Cherednik algebra of type 𝔤𝔩_{l}.

In the next two sections, we define Cherednik algebras and Yangians and explore some of their basic properties. The fourth section states the main result (theorem 4.2) for the trigonometric case, which is proved in the following one. After that, we look more closely at the action of certain elements of LY since this is useful in the last section, which concerns the rational case (theorem 7.1). Most of our results in the rational case follows from the observation that the rational Cherednik algebra of type 𝔤𝔩_{l} is contained in the trigonometric one. Furthermore, our equivalence restricts to an equivalence between two categories of BGG-type (theorem 7.2).

Acknowledgments During the preparation of this paper, the author was supported by a Pionier grant of the Netherlands Organization for Scientific Research (NWO). He warmly thanks E. Opdam for his invitation to spend the year 2005 at the University of Amsterdam. He also thanks I. Cherednik, P. Etingof, V. Ginzburg, I. Gordon and T. Nevins for their comments.

2 Hecke algebras and Cherednik algebras

The definitions given in this section could be stated for any Weyl group W. However, in this paper, we will be concerned only with the symmetric group S_{l}, so we will restrict our definitions to the case W = S_{l}. We set 𝔥 = ℂ^{l}. The symmetric group S_{l} acts on 𝔥 by permuting the coordinates; associated to 𝔥 are two polynomial algebras: ℂ[𝔥] = Sym(𝔥^{∗}) = ℂ[x_{1}, …, x_{l}] and ℂ[𝔥^{∗}] = Sym(𝔥) = ℂ[y_{1}, …, y_{l}], where {x_{1}, …, x_{l}} and {y_{1}, …, y_{l}} are dual bases of 𝔥^{∗} and 𝔥, respectively. For i ≠ j, we set α_{ij} = x_{i} − x_{j}, α_{ij}^{∨} = y_{i} − y_{j}, R = {α_{ij}|1 ≤ i ≠ j ≤ l} and R^{+} = {α_{ij}|1 ≤ i < j ≤ l}. The set Π = {x_{i} − x_{i+1}|1 ≤ i ≤ n − 1} is a basis of simple roots. The reflection in 𝔥 with respect to the hyperplane α = 0 is denoted s_{α}, so s_{α}(y) = y − ⟨α, y⟩α^{∨}, where ⟨ , ⟩ : 𝔥^{∗} × 𝔥 → ℂ is the canonical pairing. We set s_{ij} = s_{α_{ij}}.

The finite Hecke algebra H_{q} associated to S_{l} is a deformation of the group algebra ℂ[S_{l}] and the affine Hecke algebra H̃_{q} is a deformation of the group algebra of the (extended) affine Weyl group S̃_{l} = P ⋊ S_{l} where P is the lattice ⊕_{i=1}^{l}ℤx_{i} ⊂ 𝔥^{∗} (so ℂ[S̃_{l}] = ℂ[x_{1}^{±}, …, x_{l}^{±}] ⋊ S_{l}). The algebra H̃_{q} admits a degenerate form H_{c} first introduced by Drinfeld [Dr1] and by Lusztig [Lu].

Definition 2.1. The degenerate affine Hecke algebra H_{c} of type 𝔤𝔩_{l} is the algebra generated by the polynomial algebra Sym(𝔥) = ℂ[z_{1}, …, z_{l}] and the group algebra ℂ[S_{l}] with the relations

s_{α} · z − s_{α}(z)s_{α} = −c⟨α, z⟩ ∀z ∈ 𝔥, ∀α ∈ Π

Remark 2.1. The algebras H_{c_{1}} and H_{c_{2}} are isomorphic if c_{1} ≠ 0 and c_{2} ≠ 0. Clearly, H_{0} ≅ ℂ[𝔥]⋊S_{l}.

The double affine Hecke algebra ℍ introduced by I. Cherednik [Ch2] also admits degenerate versions: the trigonometric one and the rational one. Recall that the group S̃_{l} is generated by s_{α} ∀α ∈ R and by the element π = x_{1}s_{12}s_{23} ⋯ s_{l−1,l}.

Definition 2.2 (Cherednik). Let t, c ∈ ℂ. The degenerate (trigonometric) double affine Hecke algebra of type 𝔤𝔩_{l} is the algebra H_{t,c} generated by the group algebra of the (extended) affine Weyl group ℂ[S̃_{l}] and the polynomial algebra ℂ[z_{1}, …, z_{l}] = Sym(𝔥) subject to the following relations:

s_{α} · z − s_{α}(z)s_{α} = −c⟨α, z⟩ ∀z ∈ 𝔥, ∀α ∈ Π
πz_{i} = z_{i+1}π, 1 ≤ i ≤ l − 1 πz_{l} = (z_{1} − t)π

Remark 2.2. The subalgebra generated by ℂ[S_{l}] and the polynomial algebra ℂ[z_{1}, …, z_{l}] is isomorphic to the degenerate affine Hecke algebra H_{c}.

The rational version of the double affine Hecke algebra has been studied quite intensively in the past few years (see, for example, [BEG1],[GGOR]) and is usually referred to as the rational Cherednik algebra.

Definition 2.3. Let t, c ∈ ℂ. The rational Cherednik algebra H_{t,c} of type 𝔤𝔩_{l} is the algebra generated by ℂ[𝔥], ℂ[𝔥^{∗}] and ℂ[S_{l}] subject to the following relations:

w · x · w^{−1} = w(x) w · y · w^{−1} = w(y) ∀x ∈ 𝔥^{∗}, ∀y ∈ 𝔥

[y, x] = yx − xy = t⟨y, x⟩ + c ∑_{α∈R^{+}}⟨α, y⟩⟨x, α^{∨}⟩s_{α}

Remark 2.3. The rational Cherednik algebra of type A_{l−1} is the subalgebra of H_{t,c} generated by ℂ[x_{i} − x_{j}] ⊂ ℂ[x_{1}, …, x_{l}], by ℂ[y_{i} − y_{j}] ⊂ ℂ[y_{1}, …, y_{l}] and by ℂ[S_{l}].

The exists a simple relation between H_{t,c} and H_{t,c}. (See also [Su].)

Proposition 2.1. The algebra ℂ[x_{1}^{±}, …, x_{l}^{±}] ⊗_{ℂ[𝔥]} H_{t,c} is isomorphic to H_{t,c}.

Before giving a proof of this proposition, we need to introduce elements in H_{t,c} which will be very important later. For 1 ≤ i ≤ l, set U_{i} = \frac{t}{2} + x_{i}y_{i} + c ∑_{j<i} s_{ij} and Y_{i} = U_{i} + \frac{c}{2} ∑_{j≠i} sign(j − i)s_{ij} = \frac{t}{2} + x_{i}y_{i} + \frac{c}{2} ∑_{j≠i} s_{ij}.

Proposition 2.2. [DuOp],[EtGi]

1. Y_{i} = \frac{1}{2}(x_{i}y_{i} + y_{i}x_{i}).

2. U_{i}U_{j} = U_{j}U_{i} for any i, j.

3. w · Y_{i} · w^{−1} = Y_{w(i)}.

4. The elements U_{i}, 1 ≤ i ≤ l, and ℂ[S_{l}] generate a subalgebra of H_{t,c} isomorphic to the degenerate affine Hecke algebra H_{c}.

Remark 2.4. The elements Y_{i} are not pairwise commutative if c ≠ 0:

[Y_{j}, Y_{k}] = \frac{c^{2}}{4} ∑_{i≠j,k^{i=1}}^{l}[s_{ij}, s_{jk}].

Proof. The first statement follows from the equality

y_{i}x_{i} − x_{i}y_{i} = t + c ∑_{j=1,j≠i}^{l}⟨x_{i} − x_{j}, y_{i}⟩⟨x_{i}, y_{i} − y_{j}⟩s_{ij} = t + c ∑_{j=1,j≠i}^{l} s_{ij}.

The second part is proved in [DuOp]. The third part is obvious, so we prove only the fourth one. If |k − i| > 1, then s_{k,k+1}U_{i} = U_{i}s_{k,k+1}, so the non-trivial relations that we have to check involve s_{i−1,i} and s_{i,i+1}:

s_{i−1,i}U_{i} = (\frac{t}{2} + x_{i−1}y_{i−1})s_{i−1,i} + c ∑_{j<i−1} s_{i−1,j}s_{i−1,i} + c = U_{i−1}s_{i−1,i} + c

s_{i,i+1}U_{i} = (\frac{t}{2} + x_{i+1}y_{i+1})s_{i,i+1} + c ∑_{j<i+1} s_{i+1,j}s_{i+1,i} − c = U_{i+1}s_{i,i+1} − c

These two equalities, combined with the PBW-property of H_{t,c} [EtGi] and of H_{c}, complete the proof of part 4.

In the proof of the two main theorems, we will need the following identities.

Proposition 2.3. 1. If i ≠ j, then [y_{j}, x_{i}] = −cs_{ij} and [x_{i}^{−1}, y_{j}] = −cx_{i}^{−1}x_{j}^{−1}s_{ij}.

2. [y_{i}, x_{i}] = t + c ∑_{k≠i} s_{ij} and [x_{i}^{−1}, y_{i}] = tx_{i}^{−2} + c ∑_{j≠i} x_{i}^{−1}x_{j}^{−1}s_{ij}.

3. If i ≠ j, then [Y_{j}, x_{i}] = −\frac{c}{2}(x_{i} + x_{j})s_{ij} and [x_{i}^{−1}, Y_{j}] = −\frac{c}{2}(x_{i}^{−1} + x_{j}^{−1})s_{ij}.

4. [Y_{i}, x_{i}] = tx_{i} + \frac{c}{2} ∑_{j≠i}(x_{i} + x_{j})s_{ij} and [x_{i}^{−1}, Y_{i}] = tx_{i}^{−1} + \frac{c}{2} ∑_{j≠i}(x_{i}^{−1} + x_{j}^{−1})s_{ij}.

Proof. These are all immediate consequences of the definition of H_{t,c}.

Proof. (of proposition 2.1) Because of proposition 2.2, part 4, and the PBW-property of H_{t,c} and H_{t,c}, we only have to check the relation involving π in definition 2.2. First, assume that i ≠ l.

πU_{i} = (x_{1}s_{12} ⋯ s_{l−1,l})(U_{i}) = x_{1}(U_{i+1} − cs_{1,i+1})s_{12} ⋯ s_{l−1,l}
= ([x_{1}, U_{i+1}] + U_{i+1}x_{1} − cx_{1}s_{1,i+1})s_{12} ⋯ s_{l−1,l}
= (cx_{i+1}s_{1,i+1} + c[x_{1}, s_{1,i+1}] + U_{i+1}x_{1} − cx_{1}s_{1,i+1})s_{12} ⋯ s_{l−1,l}
= (U_{i+1})x_{1}s_{12} ⋯ s_{l−1,l} = U_{i+1}π

If i = l, we obtain:

πU_{l} = x_{1}(U_{1} + c ∑_{j=2}^{l} s_{1,j})s_{12} ⋯ s_{l−1,l} = (x_{1}[x_{1}, y_{1}] + cx_{1} ∑_{j=2}^{l} s_{1,j} + U_{1}x_{1})s_{12} ⋯ s_{l−1,l}

= (x_{1}(−t − c ∑_{i≠1} s_{i,1}) + cx_{1} ∑_{j=2}^{l} s_{1,j} + U_{1}x_{1})s_{12} ⋯ s_{l−1,l} = (U_{1} − t)π

Corollary 2.1 (of proposition 2.1). The algebra H_{t,c} can also be defined as the algebra generated by the elements x_{1}^{±}, …, x_{l}^{±}, Y_{1}, …, Y_{l} and S_{l} with the relations

w · x_{i} · w^{−1} = x_{w(i)} w · Y_{i} · w^{−1} = Y_{w(i)} [Y_{j}, Y_{k}] = \frac{c^{2}}{4} ∑_{i≠j,k^{i=1}}^{l}(s_{jk}s_{ik} − s_{kj}s_{ij})

x_{i}Y_{j} − Y_{j}x_{i} = tδ_{ij}x_{i} + \frac{c}{2} ∑_{α∈R^{+}}⟨α, y_{j}⟩⟨x_{i}, α^{∨}⟩(x_{i}s_{α} + s_{α}x_{i}).

3 Finite and loop Yangians

The Yangians of finite type are quantum groups, introduced by V. Drinfeld in [Dr1], which are quantizations of the enveloping algebra of the polynomial loop algebra 𝔤 ⊗_{ℂ} ℂ[u] of a semisimple Lie algebra 𝔤. The second definition in [Dr2] is given in terms of a finite Cartan matrix. If we replace it with a Cartan matrix of affine type, we obtain algebras that we call loop Yangians LY_{β,λ}. (The only occurrence of these algebras in the literature that the author could find is in a remark in [Va].) Let C = (c_{ij})_{1≤i,j≤n−1} (Ĉ = (c_{ij})_{0≤i,j≤n−1}) be a Cartan matrix of finite (resp. affine) type A_{n−1} (resp. Â_{n−1}). If n ≥ 3:

Ĉ = (\begin{matrix}2 & −1 & 0 & ⋯ & ⋯ & 0 & −1 \\ −1 & 2 & −1 & 0 & ⋯ & ⋯ & 0 \\ 0 & −1 & 2 & −1 & 0 & ⋯ & 0 \\ ⋮ & & ⋮ & & ⋮ & & ⋮ \\ 0 & ⋯ & 0 & −1 & 2 & −1 & 0 \\ 0 & ⋯ & ⋯ & 0 & −1 & 2 & −1 \\ −1 & 0 & ⋯ & ⋯ & 0 & −1 & 2 \end{matrix})

Definition 3.1. [Dr2], [ChPr2] Let λ ∈ ℂ. The Yangian Y_{λ} associated to C is the algebra generated by X_{i,r}^{±}, H_{i,r}, i = 1, … n − 1, r ∈ ℤ_{≥0}, which satisfy the following relations :

[H_{i,r}, H_{j,s}] = 0, [H_{i,0}, X_{j,s}^{±}] = ±c_{ij}X_{j,s}^{±}  (1)
[H_{i,r+1}, X_{j,s}^{±}] − [H_{i,r}, X_{j,s+1}^{±}] = ±\frac{λ}{2}c_{ij}(H_{i,r}X_{j,s}^{±} + X_{j,s}^{±}H_{i,r})  (2)
[X_{i,r}^{+}, X_{j,s}^{−}] = δ_{ij}H_{i,r+s} [X_{i,r}^{±}, X_{j,s}^{±}] = 0 if 1 < |j − i| < n − 1  (3)
[X_{i,r+1}^{±}, X_{j,s}^{±}] − [X_{i,r}^{±}, X_{j,s+1}^{±}] = ±\frac{λ}{2}c_{ij}(X_{i,r}^{±}X_{j,s}^{±} + X_{j,s}^{±}X_{i,r}^{±})  (4)
[X_{i,r_{1}}^{±}, [X_{i,r_{2}}^{±}, X_{j,s}^{±}]] + [X_{i,r_{2}}^{±}, [X_{i,r_{1}}^{±}, X_{j,s}^{±}]] = 0 ∀r_{1}, r_{2}, s ≥ 0 if j − i ≡ ±1 mod n  (5)

Remark 3.1. The Yangian Y_{λ_{1}} is isomorphic to Y_{λ_{2}} if λ_{1} ≠ 0 and λ_{2} ≠ 0.

Definition 3.2. Let β, λ ∈ ℂ. The Yangian LY_{β,λ} associated to Ĉ is the algebra generated by X_{i,r}^{±}, H_{i,r}, i = 0, … n−1, r ∈ ℤ_{≥0}, which satisfy the relations of definition 3.1 for i, j ∈ {0, …, n−1} except that the relations (2),(4) must be modified for i = 0 and j = 1, n − 1 in the following way when n ≥ 3:

[H_{1,r+1}, X_{0,s}^{±}] − [H_{1,r}, X_{0,s+1}^{±}] = (β − \frac{λ}{2} ∓ \frac{λ}{2})H_{1,r}X_{0,s}^{±} + (\frac{λ}{2} ∓ \frac{λ}{2} − β)X_{0,s}^{±}H_{1,r}  (6)

[H_{0,r+1}, X_{1,s}^{±}] − [H_{0,r}, X_{1,s+1}^{±}] = (\frac{λ}{2} ∓ \frac{λ}{2} − β)H_{0,r}X_{1,s}^{±} + (β − \frac{λ}{2} ∓ \frac{λ}{2})X_{1,s}^{±}H_{0,r}  (7)
[H_{0,r+1}, X_{n−1,s}^{±}] − [H_{0,r}, X_{n−1,s+1}^{±}] = (β − \frac{λ}{2} ∓ \frac{λ}{2})H_{0,r}X_{n−1,s}^{±} + (\frac{λ}{2} ∓ \frac{λ}{2} − β)X_{n−1,s}^{±}H_{0,r} (8)
[H_{n−1,r+1}, X_{0,s}^{±}] − [H_{n−1,r}, X_{0,s+1}^{±}] = (\frac{λ}{2} ∓ \frac{λ}{2} − β)H_{n−1,r}X_{0,s}^{±} + (β − \frac{λ}{2} ∓ \frac{λ}{2})X_{0,s}^{±}H_{n−1,r} (9)
[X_{1,r+1}^{±}, X_{0,s}^{±}] − [X_{1,r}^{±}, X_{0,s+1}^{±}] = (β − \frac{λ}{2} ∓ \frac{λ}{2})X_{1,r}^{±}X_{0,s}^{±} + (\frac{λ}{2} ∓ \frac{λ}{2} − β)X_{0,s}^{±}X_{1,r}^{±}  (10)
[X_{0,r+1}^{±}, X_{n−1,s}^{±}] − [X_{0,r}^{±}, X_{n−1,s+1}^{±}] = (β − \frac{λ}{2} ∓ \frac{λ}{2})X_{0,r}^{±}X_{n−1,s}^{±} + (\frac{λ}{2} ∓ \frac{λ}{2} − β)X_{n−1,s}^{±}X_{0,r}^{±} (11)

We will also impose the relation ∑_{i=0}^{n−1} H_{i,0} = 0.

Remark 3.2. We will write X_{i}^{±} and H_{i} instead of X_{i,0}^{±} and H_{i,0}. If β = \frac{λ}{2}, the relations defining LY_{β,λ} are the same as those in definition 3.1 with i, j ∈ {0, …, n − 1}. Note also that the relations (6),(7),(8) and (9) all follow from (10) and (11) using relation (3); they were added above as a convenient reference since they will be useful later in our computations. We should also note that LY_{β_{1},λ_{1}} ≅ LY_{β_{2},λ_{2}} if β_{2} = ηβ_{1} and λ_{2} = ηλ_{1} for some η ≠ 0.

Remark 3.3. When β = 0, LY_{λ,β} is isomorphic to the enveloping algebra of the universal central extension of 𝔰𝔩_{n}(A_{β}) where A_{β} is isomorphic to the ring of algebraic differential operators on ℂ^{∗} (see [Gu]).

Let Δ = {ϵ_{ij}, 1 ≤ i ≠ j ≤ n} be the root system of type A_{n−1}. For a positive root ϵ ∈ Δ^{+}, we denote by X_{ϵ}^{±} the corresponding standard root vector of 𝔰𝔩_{n}. If ϵ = ϵ_{ij}, i < j, then X_{ϵ}^{+} = E_{ij} and X_{ϵ}^{−} = E_{ji}, where E_{rs} is the matrix with 1 in the (r, s)-entry and zeros everywhere else. In particular, X_{θ}^{+} = E_{1n} and X_{θ}^{−} = E_{n1}, where θ is the longest root of 𝔰𝔩_{n}. If ϵ = ϵ_{i} = ϵ_{i,i+1}, then X_{ϵ}^{±} = X_{i}^{±}.

One useful observation is that these two Yangians are generated by X_{i,r}^{±}, H_{i,r}, i = 1, … n − 1 (resp. i = 0, …, n − 1) with r = 0, 1 only. The other elements are obtained inductively by the formulas:

X_{i,r+1}^{±} = ±\frac{1}{2}[H_{i,1}, X_{i,r}^{±}] − \frac{1}{2}(H_{i}X_{i,r}^{±} + X_{i,r}^{±}H_{i}), H_{i,r+1} = [X_{i,r}^{+}, X_{i,1}^{−}].  (12)

Furthermore, the subalgebra generated by the elements with r = 0 is isomorphic to the enveloping algebra of the Lie (resp. loop) algebra 𝔰𝔩_{n} (resp. L𝔰𝔩_{n} = 𝔰𝔩_{n} ⊗_{ℂ} ℂ[u, u^{−1}]). We identify X_{0}^{+} with X_{θ}^{−} ⊗ u and X_{0}^{−} with X_{θ}^{+} ⊗ u^{−1}. The subalgebra Y_{λ}^{0} generated by the elements with i ≠ 0 is a quotient of the Yangian Y_{λ}.

The two subalgebras Y_{λ}^{0} and 𝔘(L𝔰𝔩_{n}) generate LY_{β,λ}. Indeed, combining the observations in the previous two paragraphs, we see that we only have to show that the subalgebra they generate contains X_{0,1}^{±}. From the relation (1) in definition 3.2 with i = 1, we know that [H_{1}, X_{0,1}^{±}] = ∓X_{0,1}^{±}, so, substituting into equation (6), we obtain

[H_{1,1}, X_{0}^{±}] ± X_{0,1}^{±} = (β − \frac{λ}{2} ∓ \frac{λ}{2})H_{1}X_{0}^{±} + (\frac{λ}{2} ∓ \frac{λ}{2} − β)X_{0}^{±}H_{1}.

Thus X_{0,1}^{±} (hence also H_{0,1}) belongs to the subalgebra of LY_{β,λ} generated by Y_{λ}^{0} and 𝔘(L𝔰𝔩_{n}). When no confusion is possible, we will write LY and Y instead of LY_{β,λ} and Y_{λ}.

For 1 ≤ i ≤ n − 1, set

J(X_{i}^{±}) = X_{i,1}^{±} + λω_{i}^{±} where ω_{i}^{±} = ±\frac{1}{4} ∑_{ϵ∈Δ^{+}} ([X_{i}^{±}, X_{ϵ}^{±}]X_{ϵ}^{∓} + X_{ϵ}^{∓}[X_{i}^{±}, X_{ϵ}^{±}]) − \frac{1}{4}(X_{i}^{±}H_{i} + H_{i}X_{i}^{±})

J(H_{i}) = H_{i,1} + λν_{i} where ν_{i} = \frac{1}{4} ∑_{ϵ∈Δ^{+}}(ϵ, ϵ_{i})(X_{ϵ}^{+}X_{ϵ}^{−} + X_{ϵ}^{−}X_{ϵ}^{+}) − \frac{1}{2}H_{i}^{2}.

More explicitly, since X_{i}^{+} = E_{i,i+1}, X_{i}^{−} = E_{i+1,i} and H_{i} = E_{ii} − E_{i+1,i+1} for 1 ≤ i ≤ n − 1, we can write

ω_{i}^{+} = \frac{1}{4} ∑_{j≠i,i+1^{j=1}}^{n} sign(j − i)(E_{j,i+1}E_{ij} + E_{ij}E_{j,i+1}) − \frac{1}{4}(E_{i,i+1}H_{i} + H_{i}E_{i,i+1})  (13)

ω_{i}^{−} = \frac{1}{4} ∑_{j≠i,i+1^{j=1}}^{n} sign(j − i)(E_{i+1,j}E_{ji} + E_{ji}E_{i+1,j}) − \frac{1}{4}(E_{i+1,i}H_{i} + H_{i}E_{i+1,i})  (14)

It is possible to define elements J(z) ∈ Y for any z ∈ 𝔰𝔩_{n} in such a way that [J(z_{1}), z_{2}] = J([z_{1}, z_{2}]): this follows from the isomorphism given in [Dr2] between two different realizations of the Yangian Y_{λ}: the one given above and the one first given in [Dr1] in terms of generators z, J(z) ∀z ∈ 𝔰𝔩_{n} (the J(z)’s satisfy a“deformed” Jacobi identity).

In the proof of our first main theorem, the following algebra automorphism will be very important.

Lemma 3.1. It is possible to define an automorphism ρ of LY by setting

ρ(H_{i,r}) = ∑_{s=0}^{r} ( \begin{matrix}r \\ s \end{matrix} ) (\frac{λ}{2})^{r−s} H_{i−1,s}, ρ(X_{i,r}^{±}) = ∑_{s=0}^{r} ( \begin{matrix}r \\ s \end{matrix} ) (\frac{λ}{2})^{r−s} X_{i−1,s}^{±} for i ≠ 0, 1

ρ(H_{i,r}) = ∑_{s=0}^{r} ( \begin{matrix}r \\ s \end{matrix} ) β^{r−s}H_{i−1,s}, ρ(X_{i,r}^{±}) = ∑_{s=0}^{r} ( \begin{matrix}r \\ s \end{matrix} ) β^{r−s}X_{i−1,s}^{±} for i = 0, 1

We use the convention that X_{−1,r}^{±} = X_{n−1,r}^{±} and H_{−1,r} = H_{n−1,r}. Note that, in particular, ρ(X_{i}^{±}) = X_{i−1}^{±}, ρ(H_{i}) = H_{i−1} ∀i and ρ(X_{i,1}^{±}) = X_{i−1,1}^{±} + \frac{λ}{2}X_{i−1}^{±}, ρ(H_{i,1}) = H_{i−1,1} + \frac{λ}{2}H_{i−1} if i ≠ 0, 1, whereas ρ(X_{i,1}^{±}) = X_{i−1,1}^{±} + βX_{i−1}^{±}, ρ(H_{i,1}) = H_{i−1,1} + βH_{i−1} if i = 0, 1 The automorphism ρ is very similar to the automorphism τ_{\frac{λ}{2}} (or τ_{β}) in [ChPr2] followed by a decrement of the indices.

Proof of lemma 3.1. We have to verify that ρ is indeed an automorphism of LY, that is, that it respects the defining relations of LY. In the case when i, j ≠ 0, 1 in the relations (1)-(5), this follows from the fact that ρ is the same as the automorphism τ_{\frac{λ}{2}} from [ChPr2] followed by a decrement of the indices. A short verification shows that ρ preserves the relations (1),(3) and (5) when i = 0, 1 or j = 0, 1. (In the case of equation (3) and i = j, , one has to use the identity

∑_{a+b=k} ( \begin{matrix}r \\ a \end{matrix} ) ( \begin{matrix}s \\ b \end{matrix} ) = ( \begin{matrix}r + s \\ k \end{matrix} ).) Since the relations (6)-(9) follow from (10) and (11) by

applying [·, X_{?,0}^{∓}],? = 0, 1, n − 1, there are three cases left that require a more detailed verification.

We will use the identity ( \begin{matrix}r \\ a \end{matrix} ) = ( \begin{matrix}r − 1 \\ a \end{matrix} ) + ( \begin{matrix}r − 1 \\ a − 1 \end{matrix} ).

Case 1: We start with the relation (4) with i = 2, j = 1. We find that ρ([X_{2,r+1}^{±}, X_{1,s}^{±}] − [X_{2,r}^{±}, X_{1,s+1}^{±}]) is equal to

= ∑_{a=0}^{r+1} ∑_{b=0}^{s} ( \begin{matrix}r + 1 \\ a \end{matrix} ) ( \begin{matrix}s \\ b \end{matrix} ) [(\frac{λ}{2})^{r+1−a} X_{1,a}^{±}, β^{s−b}X_{0,b}^{±}]

− ∑_{a=0}^{r} ∑_{b=0}^{s+1} ( \begin{matrix}r \\ a \end{matrix} ) ( \begin{matrix}s + 1 \\ b \end{matrix} ) [(\frac{λ}{2})^{r−a} X_{1,a}^{±}, β^{s+1−b}X_{0,b}^{±}]

= ∑_{a=0}^{r+1} ∑_{b=0}^{s} (( \begin{matrix}r \\ a \end{matrix} ) + ( \begin{matrix}r \\ a − 1 \end{matrix} )) ( \begin{matrix}s \\ b \end{matrix} ) (\frac{λ}{2})^{r+1−a} β^{s−b}[X_{1,a}^{±}, X_{0,b}^{±}]

− ∑_{a=0}^{r} ∑_{b=0}^{s+1} ( \begin{matrix}r \\ a \end{matrix} ) (( \begin{matrix}s \\ b \end{matrix} ) + ( \begin{matrix}s \\ b − 1 \end{matrix} )) (\frac{λ}{2})^{r−a} β^{s+1−b}[X_{1,a}^{±}, X_{0,b}^{±}]

= ∑_{a=0}^{r} ∑_{b=0}^{s} ( \begin{matrix}r \\ a \end{matrix} ) ( \begin{matrix}s \\ b \end{matrix} ) (\frac{λ}{2})^{r−a} β^{s−b}(\frac{λ}{2} − β)[X_{1,a}^{±}, X_{0,b}^{±}]

+ ∑_{a=0}^{r+1} ∑_{b=0}^{s} ( \begin{matrix}r \\ a − 1 \end{matrix} ) ( \begin{matrix}s \\ b \end{matrix} ) (\frac{λ}{2})^{r−a+1} β^{s−b}[X_{1,a}^{±}, X_{0,b}^{±}]

− ∑_{a=0}^{r} ∑_{b=0}^{s+1} ( \begin{matrix}r \\ a \end{matrix} ) ( \begin{matrix}s \\ b − 1 \end{matrix} ) (\frac{λ}{2})^{r−a} β^{s−b+1}[X_{1,a}^{±}, X_{0,b}^{±}]

= ∑_{a=0}^{r} ∑_{b=0}^{s} ( \begin{matrix}r \\ a \end{matrix} ) ( \begin{matrix}s \\ b \end{matrix} ) (\frac{λ}{2})^{r−a} β^{s−b}(\frac{λ}{2} − β)[X_{1,a}^{±}, X_{0,b}^{±}]

+ ∑_{ã=0}^{r} ∑_{b=0}^{s} ( r ; ã ) ( \begin{matrix}s \\ b \end{matrix} ) (\frac{λ}{2})^{r−ã} β^{s−b}[X_{1,ã+1}^{±}, X_{0,b}^{±}]

− ∑_{a=0}^{r} ∑_{b̃=0}^{s} ( \begin{matrix}r \\ a \end{matrix} ) ( s ; b̃ ) (\frac{λ}{2})^{r−a} β^{s−b̃}[X_{1,a}^{±}, X_{0,b̃+1}^{±}]

= ∑_{a=0}^{r} ∑_{b=0}^{s} ( \begin{matrix}r \\ a \end{matrix} ) ( \begin{matrix}s \\ b \end{matrix} ) (\frac{λ}{2})^{r−a} β^{s−b}(\frac{λ}{2} − β)[X_{1,a}^{±}, X_{0,b}^{±}]

+ ∑_{ã=0}^{r} ∑_{b̃=0}^{s} ( r ; ã ) ( s ; b̃ ) (\frac{λ}{2})^{r−ã} β^{s−b̃}([X_{1,ã+1}^{±}, X_{0,b̃}^{±}] − [X_{1,ã}^{±}, X_{0,b̃+1}^{±}])

= ∑_{ã=0}^{r} ∑_{b̃=0}^{s} ( r ; ã ) ( s ; b̃ ) (\frac{λ}{2})^{r−ã} β^{s−b̃}(\frac{λ}{2} − β)[X_{1,ã}^{±}, X_{0,b̃}^{±}]

+ ∑_{ã=0}^{r} ∑_{b̃=0}^{s} ( r ; ã ) ( s ; b̃ ) (\frac{λ}{2})^{r−ã} β^{s−b̃}((β − \frac{λ}{2} ∓ \frac{λ}{2})X_{1,ã}^{±}X_{0,b̃}^{±} + (\frac{λ}{2} ∓ \frac{λ}{2} − β)X_{0,b̃}^{±}X_{1,ã}^{±}))

= ∑_{ã=0}^{r} ∑_{b̃=0}^{s} ( r ; ã ) ( s ; b̃ ) (\frac{λ}{2})^{r−ã} β^{s−b̃} (∓\frac{λ}{2}) (X_{1,ã}^{±}X_{0,b̃}^{±} + X_{0,b̃}^{±}X_{1,ã}^{±})

= ρ ((∓\frac{λ}{2}) (X_{2,r}^{±}X_{1,s}^{±} + X_{1,s}^{±}X_{2,r}^{±}))

Case 2: i = 1, j = 0. We have to prove that

ρ([X_{1,r+1}^{±}, X_{0,s}^{±}] − [X_{1,r}^{±}, X_{0,s+1}^{±}]) = ρ((β − \frac{λ}{2} ∓ \frac{λ}{2})X_{1,r}^{±}X_{0,s}^{±} + (\frac{λ}{2} ∓ \frac{λ}{2} − β)X_{0,s}^{±}X_{1,r}^{±}).

This case is analogous to case 1, but a little bit simpler since the term (\frac{λ}{2} − β) above becomes (β − β) = 0.

Case 3: i = 0, j = n − 1. We have to show that

ρ([X_{0,r+1}^{±}, X_{n−1,s}^{±}] − [X_{0,r}^{±}, X_{n−1,s+1}^{±}]) = ρ((β − \frac{λ}{2} ∓ \frac{λ}{2})X_{0,r}^{±}X_{n−1,s}^{±} + (\frac{λ}{2} ∓ \frac{λ}{2} − β)X_{n−1,s}^{±}X_{0,r}^{±}).

The computations are again very similar to those of case 1: the main difference is that the factor (\frac{λ}{2} − β) gets replaced by (β − \frac{λ}{2}).

4 Schur-Weyl duality functor

The Schur-Weyl duality established by M. Varagnolo and E. Vasserot [VaVa] involves, on one side, a toroidal quantum algebra (a quantized version of the enveloping algebra of the enveloping algebra of the universal central extension of the double-loop algebra 𝔰𝔩_{n}[s^{±}, t^{±}]) and, on the other side, a double affine Hecke algebra for S_{l}. Theorem 4.2 establishes a similar type of duality between the trigonometric DAHA H_{t,c} and the loop Yangian LY_{β,λ}, which extends the duality for the Yangian of finite type due to V. Drinfeld [Dr1].

Before stating the more classical results on the theme of Schur-Weyl duality, we have to define the notion of module of level l over 𝔰𝔩_{n} and the quantized enveloping algebra 𝔘_{q}𝔰𝔩_{n}. Fix a positive integer n and set V = ℂ^{n}.

Definition 4.1. A finite dimensional representation of 𝔰𝔩_{n} or 𝔘_{q}𝔰𝔩_{n} (q not a root of unity) is of level l if each of its irreducible components is isomorphic to a direct summand of V^{⊗l}.

Theorem 4.1. [Ji, Dr1, ChPr1] Fix l ≥ 1, n ≥ 2 and assume that q ∈ ℂ^{×} is not a root of unity. Let A be one of the algebras ℂ[S_{l}], H_{q}(S_{l}), H_{1}(S_{l}), H̃_{q}(S_{l}), and let B be the corresponding one (in the same order) among 𝔘𝔰𝔩_{n}, 𝔘_{q}𝔰𝔩_{n}, Y (𝔰𝔩_{n}), 𝔘_{q}𝔰̂𝔩_{n}. There exists a functor F from the category of finite dimensional right A-modules to the category of finite dimensional left B-modules which are of level l as 𝔰𝔩_{n}-modules in the first and third case (and as 𝔘_{q}𝔰𝔩_{n}-modules in the second and fourth case) which is given by

F(M) = M ⊗_{C} V^{⊗l}

where C = ℂ[S_{l}] (first and third case) or C = H_{q}(S_{l}) (second and fourth case). Furthermore, this functor is an equivalence of categories if l ≤ n − 1.

The 𝔰𝔩_{n} module structure on V^{⊗l} commutes with the S_{l}-module structure obtained by simply permuting the factors in the tensor product. Let M be a right module over H_{t,c}. Since ℂ[S_{l}] ⊂ H_{t,c}, we can form the tensor product F(M) = M ⊗_{ℂ[S_{l}]} V^{⊗l}.

On one hand, since H_{t,c} contains the degenerate affine Hecke algebra H_{c}, M can be viewed as a right module over H_{c}, so it follows from [Dr1] that F(M) is a module of level l over the Yangian Y_{λ} of 𝔰𝔩_{n} with λ = c. On the other hand, H_{t,c} also contains a copy of the group algebra of the extended affine Weyl group S̃_{l}, so it follows from [ChPr1] (the case q = 1) that F(M) is also a module of level l over the loop algebra L𝔰𝔩_{n}. These two module structures can be glued together to obtain a module over LY. This is the content of our first main theorem. Before stating it, we need one definition.

Definition 4.2. A module M over L𝔰𝔩_{n} is called integrable if it is the direct sum of its integral weight spaces under the action of 𝔥 and if each generator X_{i,r}^{±} acts locally nilpotently on M.

Theorem 4.2. Suppose that l ≥ 1, n ≥ 3 and set λ = c, β = \frac{t}{2} − \frac{nc}{4} + \frac{c}{2}. The functor F : M ↦ M ⊗_{ℂ[S_{l}]} V^{⊗l} sends a right H_{t,c}-module to an integrable LY_{β,λ}-module of level l (as 𝔰𝔩_{n}-module). Furthermore, if l + 2 < n, this functor is an equivalence.

Remark 4.1. This theorem is very similar to the main result of [VaVa] where H_{t,c} is replaced by a double affine Hecke algebra and LY_{β,λ} is replaced by a toroidal quantum algebra, under the assumption that the parameter q is not a root of unity.

5 Proof of theorem 4.2

The proof of theorem 4.2 consists of two parts. First, we show how to obtain an integrable LY_{β,λ}- module structure on F(M), and then we prove that any integrable representation of LY of level l is of the form F(M). If there is no confusion possible for the values of the parameters, we will write H, H, H, LY instead of H_{t,c}, H_{c}, H_{t,c}, LY_{β,λ}.

5.1 Proof of theorem 4.2, part 1

Fix m ∈ M, v = v_{i_{1}} ⊗⋯⊗v_{i_{l}} ∈ V^{⊗l}, where {v_{1}, …, v_{n}} is the standard basis of ℂ^{n} and 1 ≤ i_{j} ≤ n. The subalgebra 𝔰𝔩_{n} generated by the elements X_{i}^{±}, H_{i}, 1 ≤ i ≤ n − 1, acts on V^{⊗l} as usual. The element z ⊗ u^{±} ∈ L𝔰𝔩_{n} acts on F(M) in the following way:

(z ⊗ u^{±k})(m ⊗ v) = ∑_{j=1}^{l} mx_{j}^{±k} ⊗ v_{i_{1}} ⊗ ⋯ ⊗ (zv_{i_{j}}) ⊗ · ⊗ v_{i_{l}}.

For z ∈ 𝔰𝔩_{n}, we will write z^{j}(v) for v_{i_{1}} ⊗ ⋯ ⊗ (zv_{i_{j}}) ⊗ · ⊗ v_{i_{l}}. The elements J(X_{i}^{±}), J(H_{i}) and X_{i,1}^{±}, H_{i,1}, 1 ≤ i ≤ n − 1, act on F(M) in the following way (see [Dr1],[ChPr2]):

J(X_{i}^{±})(m ⊗ v) = ∑_{j=1}^{l} mY_{j} ⊗ X_{i}^{±,j}(v), X_{i,1}^{±}(m ⊗ v) = J(X_{i}^{±})(m ⊗ v) − λω_{i}^{±}(m ⊗ v),

J(H_{i})(m ⊗ v) = ∑_{j=1}^{l} mY_{j} ⊗ H_{i}^{j}(v), H_{i,1}(m ⊗ v) = J(H_{i})(m ⊗ v) − λν_{i}(m ⊗ v).

Following one of the main ideas in [VaVa], we define a linear automorphism T of M ⊗_{ℂ[S_{l}]} V^{⊗l} in the following way:

T (m ⊗ v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{l}}) = (mx_{1}^{−δ_{i_{1},n}} ⋯ x_{l}^{−δ_{i_{l},n}}) ⊗ v_{i_{1}+1} ⊗ ⋯ ⊗ v_{i_{l}+1},

with the convention that v_{n+1} = v_{1}. (Here, δ_{ij} is the usual delta function.) One can check that T ∘ φ(X_{i−1}^{±}) = φ(X_{i}^{±}) ∘ T and T ∘ φ(H_{i−1}) = φ(H_{i}) ∘ T for any 0 ≤ i ≤ n − 1, where φ : Y ⟶ End_{ℂ}(F(M)) is the algebra map coming from the Y -module structure on F(M). Recall the automorphism ρ from section 3.

The following lemma will be crucial.

Lemma 5.1. Let M be a module over H. For any 2 ≤ i ≤ n − 1 and any r ≥ 0, the following identities between operators on F(M) hold:

φ(ρ(X_{i,r}^{±})) = T^{−1} ∘ φ(X_{i,r}^{±}) ∘ T φ(ρ(H_{i,r})) = T^{−1} ∘ φ(H_{i,r}) ∘ T  (15)
φ(ρ^{2}(X_{1,r}^{±})) = T^{−2} ∘ φ(X_{1,r}^{±}) ∘ T^{2} φ(ρ^{2}(H_{1,r})) = T^{−2} ∘ φ(H_{1,r}) ∘ T^{2}  (16)

Proof. Since the elements X_{i,r}^{±}, H_{i,r} with r = 0, 1 generate LY (see equation (12)), it is enough to prove the lemma for r = 0, 1. First, we prove relation (15) for X_{i,1}^{+} with 2 ≤ i ≤ n − 1. The proof for X_{i,1}^{−} is exactly the same and we omit it, and the proof for H_{i,1} follows from either of these two cases using identity (3). The following observation will be used repeatedly throughout this article: the action of s_{jk} on V^{⊗l} if given in terms of matrices by: s_{jk} = ∑_{r,s=1}^{n} E_{rs}^{j}E_{sr}^{k}.

Recall that X_{i,1}^{+} = J(X_{i}^{+}) − λω_{i}^{+}. To simplify the notation, will not use φ in the proof. (We used it only to state the lemma in a convenient way.) We have to check the equality

(J(X_{i}^{+}) − λω_{i}^{+})(T (m ⊗ v)) = T ((J(X_{i−1}^{+}) − λω_{i−1}^{+} + \frac{λ}{2}X_{i−1}^{+})(m ⊗ v)).  (17)

With v as before, suppose that j_{1} < ⋯ < j_{p} are exactly the values of j for which i_{j} = n. Then T (m ⊗ v) = mx_{j_{1}}^{−1} ⋯ x_{j_{p}}^{−1} ⊗ v_{+1} where v_{+1} = v_{i_{1}+1} ⊗ ⋯ ⊗ v_{i_{l}+1}. We will use the abbreviation x_{j_{1},...,j_{p}}^{−1} for the product x_{j_{1}}^{−1} ⋯ x_{j_{p}}^{−1}.

J(X_{i}^{+})(T (m ⊗ v)) = J(X_{i}^{+})(mx_{j_{1},...,j_{p}}^{−1} ⊗ v_{+1}) = ∑_{k=1}^{l} mx_{j_{1},...,j_{p}}^{−1}Y_{k} ⊗ E_{i,i+1}^{k}(v_{+1})

= ∑_{k=1}^{l} ∑_{r=1}^{p} mx_{j_{1}}^{−1} ⋯ x_{j_{r−1}}^{−1}[x_{j_{r}}^{−1}, Y_{k}]x_{j_{r+1}}^{−1} ⋯ x_{j_{p}}^{−1} ⊗ E_{i,i+1}^{k}(v_{+1})  (18)

+ ∑_{k=1}^{l} mY_{k}x_{j_{1},...,j_{p}}^{−1} ⊗ E_{i,i+1}^{k}(v_{+1})  (19)

Note that E_{i,i+1}(v_{+1}) = (E_{i−1,i}(v))_{+1} since we are assuming that 2 ≤ i ≤ n − 1.

∑_{k=1}^{l} mY_{k}x_{j_{1},...,j_{p}}^{−1} ⊗ E_{i,i+1}^{k}(v_{+1}) = ∑_{k=1}^{l} mY_{k}x_{j_{1},...,j_{p}}^{−1} ⊗ (E_{i−1,i}^{k}(v))_{+1} = T (J(X_{i−1}^{+})(m ⊗ v)) (20)

Therefore, it follows from (18), (19) and (20) that the identity (17) that we must prove can be written as

(18) − λω_{i}^{+}(T (m ⊗ v)) = −λT (ω_{i−1}^{+}(m ⊗ v)) + \frac{λ}{2}T (X_{i−1}^{+}(m ⊗ v))  (21)

In the sum (18), note that if k = j_{s} for some s, then E_{i,i+1}^{k}(v_{+1}) = 0 since v_{i_{k}+1} = v_{i_{js}+1} = v_{1} and E_{i,i+1}(v_{1}) = 0. Therefore, using proposition 2.3, we can simplify expression (18):

(18) = ∑_{r=1}^{p} ∑_{k≠j_{s} ∀s^{k=1}}^{l} mx_{j_{1}}^{−1} ⋯ x_{j_{r−1}}^{−1} (−\frac{c}{2}) (x_{j_{r}}^{−1} + x_{k}^{−1})s_{k,j_{r}}x_{j_{r+1}}^{−1} ⋯ x_{j_{p}}^{−1} ⊗ E_{i,i+1}^{k}(v_{+1})

(18) = −\frac{c}{2} ∑_{r=1}^{p} ∑_{k≠j_{s}∀s^{k=1}}^{l} mx_{j_{1}}^{−1} ⋯ x_{j_{r−1}}^{−1}x_{j_{r}}^{−1}x_{j_{r+1}}^{−1} ⋯ x_{j_{p}}^{−1} ⊗ E_{i,i+1}^{j_{r}}(s_{k,j_{r}}(v_{+1}))  (22)

−\frac{c}{2} ∑_{r=1}^{p} ∑_{k≠j_{s}∀s^{k=1}}^{l} mx_{j_{1}}^{−1} ⋯ x_{j_{r−1}}^{−1}x_{k}^{−1}x_{j_{r+1}}^{−1} ⋯ x_{j_{p}}^{−1} ⊗ E_{i,i+1}^{j_{r}}(s_{k,j_{r}}(v_{+1}))  (23)

We now turn to the second term on the left-hand side in (21). We distinguish two cases: when E_{j,i+1} and E_{ij} act on the same tensorand, and when they act on different ones.

λω_{i}^{+}(T (m ⊗ v)) = \frac{λ}{4} ∑_{j≠i,i+1^{j=1}}^{n} ∑_{k=1}^{l} sign(j − i)mx_{j_{1},...,j_{p}}^{−1} ⊗ E_{i,i+1}^{k}(v_{+1})  (24)

+\frac{λ}{2} ∑_{j≠i,i+1^{j=1}}^{n} ∑_{i_{k}=j−1^{k=1}}^{l} ∑_{i_{d}=i^{d=1}}^{l} sign(j − i)mx_{j_{1},...,j_{p}}^{−1} ⊗ E_{i,i+1}^{k}(s_{kd}(v_{+1})) (25)

−\frac{λ}{4}mx_{j_{1},...,j_{p}}^{−1} ⊗ (E_{i,i+1}H_{i} + H_{i}E_{i,i+1})(v_{+1})  (26)

The term (24) can be simplified:

(24) = \frac{λ}{4}(n − 2i) ∑_{k=1}^{l} mx_{j_{1},...,j_{p}}^{−1} ⊗ E_{i,i+1}^{k}(v_{+1}) = λ (\frac{n − 2i}{4}) T (∑_{k=1}^{l} m ⊗ E_{i−1,i}^{k}(v))  (27)

We now consider λT (ω_{i−1}^{+}(m ⊗ v)). As for ω_{i}^{+} above, we distinguish two cases.

λT (ω_{i−1}^{+}(m ⊗ v)) = T (\frac{λ}{4} ∑_{j≠i−1,i^{j=1}}^{n} ∑_{k=1}^{l} sign(j − i + 1)m ⊗ E_{i−1,i}^{k}(v))  (28)

+T (\frac{λ}{2} ∑_{j≠i−1,i^{j=1}}^{n} ∑_{i_{k}=j^{k=1}}^{l} ∑_{i_{d}=i^{d=1}}^{l} sign(j − i + 1)m ⊗ E_{i−1,i}^{k}(s_{kd}(v)))  (29)

−\frac{λ}{4}T ((E_{i−1,i}H_{i−1} + H_{i−1}E_{i−1,i})(m ⊗ v))  (30)

The expressions (26) and (30) are identical and the difference between the sums (28) and (24) is equal to \frac{λ}{2}T (∑_{k=1}^{l} m ⊗ E_{i−1,i}^{k}(v)) (see (27)), that is, \frac{λ}{2}T (X_{i−1}^{+}(m ⊗ v)). Therefore, the equality (21) that we have to prove simplifies to (18) = (25) − (29).

In expression (25), we consider two different cases: j ≠ 1 and j = 1 (hence i_{k} = n):

(25) = \frac{λ}{2} ∑_{j≠i,i+1^{j=2}}^{n} ∑_{i_{k}=j−1^{k=1}}^{l} ∑_{i_{d}=i^{d=1}}^{l} sign(j − i)mx_{j_{1},...,j_{p}}^{−1} ⊗ (E_{i−1,i}^{k}(s_{kd}(v)))_{+1}  (31)

−\frac{λ}{2} ∑_{i_{k}=n^{k=1}}^{l} ∑_{i_{d}=i^{d=1}}^{l} mx_{j_{1},...,j_{p}}^{−1} ⊗ E_{i,i+1}^{k}(s_{kd}(v_{+1}))  (32)

We can also decompose expression (29) into two sums by considering separately the cases j ≠ n and j = n.

(29) = \frac{λ}{2}T ( ∑_{j≠i−1,i^{j=1}}^{n−1} ∑_{i_{k}=j^{k=1}}^{l} ∑_{i_{d}=i^{d=1}}^{l} sign(j − i)m ⊗ E_{i−1,i}^{k}(s_{kd}(v)))  (33)

+\frac{λ}{2}T ( ∑_{i_{k}=n^{k=1}}^{l} ∑_{i_{d}=i^{d=1}}^{l} m ⊗ E_{i−1,i}^{k}(s_{kd}(v)))  (34)

Note that the sums (31) and (33) are equal, so the difference (25) − (29) is equal to (32) − (34). The equality (18) = (25) − (29) is now a consequence of the observation that the sums (22) and (32) are identical, and so are (23) and −(34). (Indeed, if i_{k} = n, then k = j_{r} for some r and we have to interchange the roles of d and k to see that these expressions are indeed equal.) We also need our hypothesis that λ = c.

We now consider the case i = 1 in our proof of lemma 5.1. The identity (13) gives the following expression for ω_{1}^{+} and ω_{n−1}^{+}:

ω_{1}^{+} = \frac{1}{4} ∑_{j=3}^{n}(E_{j2}E_{1j} + E_{1j}E_{j2}) − \frac{1}{4}(E_{12}H_{1} + H_{1}E_{12})

ω_{n−1}^{+} = −\frac{1}{4} ∑_{j=1}^{n−2}(E_{jn}E_{n−1,j} + E_{n−1,j}E_{jn}) − \frac{1}{4}(E_{n−1,n}H_{n−1} + H_{n−1}E_{n−1,n})

Suppose that j_{1}, …, j_{p} (resp. γ_{1}, …, γ_{e}) are exactly the values of j (resp. of γ) such that i_{j} = n (resp. i_{γ} = n − 1). Then T^{2}(m ⊗ v) = mx_{j_{1}}^{−1} ⋯ x_{j_{p}}^{−1}x_{γ_{1}}^{−1} ⋯ x_{γ_{e}}^{−1} ⊗ v_{+2}. Since X_{1,1}^{+} = J(X_{1}^{+}) − λω_{1}^{+}, we obtain:

X_{1,1}^{+}(T^{2}(m ⊗ v)) = ∑_{k=1}^{l} mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1}Y_{k} ⊗ E_{12}^{k}(v_{+2})  (35)

−\frac{λ}{4} ∑_{j=3}^{n} mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ (E_{j2}E_{1j} + E_{1j}E_{j2})(v_{+2})  (36)

+\frac{λ}{4}mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ (E_{12}H_{1} + H_{1}E_{12})(v_{+2})  (37)

In the summation (35), we can assume that k = j_{s} for some s, since otherwise E_{12}^{k}(v_{+2}) = 0.

(35) = ∑_{s=1}^{p} ∑_{u=1}^{e} mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1}}^{−1} ⋯ x_{γ_{u−1}}^{−1}[x_{γ_{u}}^{−1}, Y_{j_{s}}]x_{γ_{u+1}}^{−1} ⋯ x_{γ_{e}}^{−1} ⊗ E_{12}^{j_{s}}(v_{+2})  (38)

+ ∑_{s=1}^{p} ∑_{r=1}^{p} mx_{j_{1}}^{−1} ⋯ x_{j_{r−1}}^{−1}[x_{j_{r}}^{−1}, Y_{j_{s}}]x_{j_{r+1}}^{−1} ⋯ x_{j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{j_{s}}(v_{+2})  (39)

+ ∑_{k=1}^{l} mY_{k}x_{j_{1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{k}(v_{+2})  (40)

The last term is equal to T^{2}(J(X_{n−1}^{+})(m⊗v)). The term (37) above is equal to \frac{λ}{4}T^{2}((E_{n−1,n}H_{n−1}+ H_{n−1}E_{n−1,n})(m ⊗ v)). Therefore,

X_{1,1}^{+}(T^{2}(m ⊗ v)) = (38) + (39) + (36) + T^{2}(X_{n−1,1}^{+}(m ⊗ v)) + (50).

(The expression (50) appears explicitly below.)

We need to decompose the sums (38) and (39).

(38) = ∑_{s=1}^{p} ∑_{u=1}^{e} mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1}}^{−1} ⋯ x_{γ_{u−1}}^{−1} (−\frac{c}{2}) (x_{γ_{u}}^{−1} + x_{j_{s}}^{−1})s_{γ_{u},j_{s}}x_{γ_{u+1}}^{−1} ⋯ x_{γ_{e}}^{−1} ⊗ E_{12}^{j_{s}}(v_{+2})

= −\frac{c}{2} ∑_{s=1}^{p} ∑_{u=1}^{e} mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1}}^{−1} ⋯ x_{γ_{u−1}}^{−1}x_{γ_{u}}^{−1}x_{γ_{u+1}}^{−1} ⋯ x_{γ_{e}}^{−1} ⊗ E_{12}^{γ_{u}}(s_{γ_{u},j_{s}}(v_{+2}))  (41)

−\frac{c}{2} ∑_{s=1}^{p} ∑_{u=1}^{e} mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1}}^{−1} ⋯ x_{γ_{u−1}}^{−1}x_{j_{s}}^{−1}x_{γ_{u+1}}^{−1} ⋯ x_{γ_{e}}^{−1} ⊗ E_{12}^{γ_{u}}(s_{γ_{u},j_{s}}(v_{+2}))  (42)

(39) = ∑_{a=1}^{p} ∑_{d≠a^{d=1}}^{p} mx_{j_{1}}^{−1} ⋯ x_{j_{d−1}}^{−1}[x_{j_{d}}^{−1}, Y_{j_{a}}]x_{j_{d+1}}^{−1} ⋯ x_{j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{j_{a}}(v_{+2})

+ ∑_{a=1}^{p} mx_{j_{1}}^{−1} ⋯ x_{j_{a−1}}^{−1}[x_{j_{a}}^{−1}, Y_{j_{a}}]x_{j_{a+1}}^{−1} ⋯ x_{j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{j_{a}}(v_{+2})

= ∑_{a=1}^{p} ∑_{d≠a^{d=1}}^{p} mx_{j_{1},...,j_{d−1}}^{−1} (−\frac{c}{2}) (x_{j_{d}}^{−1} + x_{j_{a}}^{−1})s_{j_{d},j_{a}}x_{j_{d+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{j_{a}}(v_{+2})

+ ∑_{a=1}^{p} mx_{j_{1},...,j_{a−1}}^{−1} (tx_{j_{a}}^{−1} + \frac{c}{2} ∑_{q≠j_{a}}(x_{j_{a}}^{−1} + x_{q}^{−1})s_{j_{a},q}) x_{j_{a+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{j_{a}}(v_{+2})

= −\frac{c}{2} ∑_{a=1}^{p} ∑_{d≠a^{d=1}}^{p} mx_{j_{1},...,j_{d−1}}^{−1}x_{j_{d}}^{−1}s_{j_{a},j_{d}}(x_{j_{d+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1}) ⊗ E_{12}^{j_{d}}(s_{j_{a},j_{d}}(v_{+2}))  (43)

−\frac{c}{2} ∑_{a=1}^{p} ∑_{d≠a^{d=1}}^{p} mx_{j_{1},...,j_{d−1}}^{−1}x_{j_{a}}^{−1}s_{j_{d},j_{a}}(x_{j_{d+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1}) ⊗ E_{12}^{j_{d}}(s_{j_{a},j_{d}}(v_{+2}))  (44)

+t ∑_{a=1}^{p} mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{j_{a}}(v_{+2})  (45)

+\frac{c}{2} ∑_{a=1}^{p} ∑_{q≠j_{a}^{q=1}}^{l} mx_{j_{1},...,j_{a−1}}^{−1}x_{j_{a}}^{−1}s_{j_{a},q}(x_{j_{a+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1}) ⊗ E_{12}^{q}(s_{j_{a},q}(v_{+2}))  (46)

+\frac{c}{2} ∑_{a=1}^{p} ∑_{q≠j_{a}^{q=1}}^{l} mx_{j_{1},...,j_{a−1}}^{−1}x_{q}^{−1}s_{j_{a},q}(x_{j_{a+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1}) ⊗ E_{12}^{q}(s_{j_{a},q}(v_{+2}))  (47)

We now focus on ω_{1}^{+}(T^{2}(m ⊗ v)) and T^{2}(ω_{n−1}^{+}(m ⊗ v)). We’ve used the equality ω_{1}^{+}(T^{2}(m ⊗ v)) = −(36) − (37) earlier. We can decompose (36) by considering the cases when E_{1j} and E_{j2} act on the same tensorand and on different ones:

−(36) = λ (\frac{n − 2}{4}) mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}(v_{+2})  (48)

+\frac{λ}{2} ∑_{j=3}^{n} ∑_{i_{q}+2=j^{q=1}}^{l} ∑_{b=1}^{p} mx_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{q}(s_{q,j_{b}}(v_{+2}))  (49)

λT^{2}(ω_{n−1}^{+}(m ⊗ v)) = −\frac{λ}{4}T^{2} (∑_{j=1}^{n−2}(E_{jn}E_{n−1,j} + E_{n−1,j}E_{jn})(m ⊗ v))  (50)

−\frac{λ}{4}T^{2}((E_{n−1,n}H_{n−1} + H_{n−1}E_{n−1,n})(m ⊗ v))  (51)

We observe that −(37) = (51). As with (36), we can decompose (50):

(50) = −λ (\frac{n − 2}{4}) mx_{j_{1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ (E_{n−1,n}(v))_{+2}  (52)

−\frac{λ}{2}T^{2} (∑_{j=1}^{n−2} ∑_{i_{q}=j^{q=1}}^{l} ∑_{b=1}^{p} m ⊗ E_{n−1,n}^{q}(s_{q,j_{b}}(v)))  (53)

We observe that (48) − (52) = λ (\frac{n−2}{2}) T^{2}(E_{n−1,n}(m ⊗ v)).

(53) = −\frac{λ}{2} ∑_{j=1}^{n−2} ∑_{i_{q}=j^{q=1}}^{l} ∑_{b=1}^{p} mx_{j_{1},...,j_{p}}^{−1} \frac{x_{j_{b}}}{x_{q}} x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{q}(s_{q,j_{b}}(v_{+2}))

To obtain the last expression, note that i_{q} ≠ j_{a}, γ_{h} for any a, h since i_{q} = j and we consider values of j different from n and n − 1.

We now decompose the sums (46) and (47) into three different sums. In the first case, q = j_{d} ≠ j_{a}; in the second one, q = γ_{h}; and in the third case q ≠ j_{a}, γ_{h} for any a, h.

(46) = \frac{c}{2} ∑_{a=1}^{p} ∑_{d≠a^{d=1}}^{p} mx_{j_{1},...,j_{a−1}}^{−1}x_{j_{a}}^{−1}s_{j_{a},j_{d}}(x_{j_{a+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1}) ⊗ E_{12}^{j_{d}}(s_{j_{a},j_{d}}(v_{+2}))  (54)

+\frac{c}{2} ∑_{a=1}^{p} ∑_{h=1}^{e} mx_{j_{1},...,j_{a−1}}^{−1}x_{j_{a}}^{−1}x_{j_{a+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{h−1}}^{−1}x_{j_{a}}^{−1}x_{γ_{h+1},...,γ_{e}}^{−1} ⊗ E_{12}^{γ_{h}}(s_{j_{a},γ_{h}}(v_{+2})) (55)

+\frac{c}{2} ∑_{a=1}^{p} ∑_{q≠j_{d},γ_{h}^{q=1}}^{l} mx_{j_{1},...,j_{a−1}}^{−1}x_{j_{a}}^{−1}x_{j_{a+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{q}(s_{j_{a},q}(v_{+2}))  (56)

(47) = \frac{c}{2} ∑_{a=1}^{p} ∑_{d≠a^{d=1}}^{p} mx_{j_{1},...,j_{a−1}}^{−1}x_{j_{d}}^{−1}s_{j_{a},j_{d}}(x_{j_{a+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1}) ⊗ E_{12}^{j_{d}}(s_{j_{a},j_{d}}(v_{+2}))  (57)

+\frac{c}{2} ∑_{a=1}^{p} ∑_{h=1}^{e} mx_{j_{1},...,j_{a−1}}^{−1}x_{γ_{h}}^{−1}x_{j_{a+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{h−1}}^{−1}x_{j_{a}}^{−1}x_{γ_{h+1},...,γ_{e}}^{−1} ⊗ E_{12}^{γ_{h}}(s_{j_{a},γ_{h}}(v_{+2})) (58)

+\frac{c}{2} ∑_{a=1}^{p} ∑_{q≠j_{d},γ_{h}^{q=1}}^{l} mx_{j_{1},...,j_{a−1}}^{−1}x_{q}^{−1}x_{j_{a+1},...,j_{p}}^{−1}x_{γ_{1},...,γ_{e}}^{−1} ⊗ E_{12}^{q}(s_{j_{a},q}(v_{+2}))  (59)

The following equalities hold since we are assuming that λ = c:

(37) = −(51), (56) = (49), (59) = −(53), (58) = −(41), (42) = −(55), (43) = −(57), (44) = −(54)

Using our assumption that β = \frac{t}{2} − \frac{λ(n−2)}{4}, we can prove that X_{1,1}^{+}(T^{2}(m ⊗ v)) = T^{2}((X_{n−1,1}^{+} + 2βX_{n−1}^{+})(m ⊗ v)):

X_{1,1}^{+}(T^{2}(m ⊗ v)) − T^{2}(X_{n−1,1}^{+}(m ⊗ v)) = (35) + (36) + (37)
= (38) + (39) + (41) + T^{2}(J(X_{n−1}^{+})(m ⊗ v)) − T^{2}(X_{n−1,1}^{+}(m ⊗ v))
= (41) + … + (47) + (36) + (37) + (50) + (51)
= (41) + … (45) + (54) + (55) + (56) + (57) + (58) + (59) − (48) − (49)

+(37) + (51) + (52) + (53)

= ((37) + (51)) + ((41) + (58)) + ((42) + (55)) + ((43) + (57)) + ((44) + (54)) + (45) − (48)

+(−(49) + (56)) + ((53) + (59)) + (52)

= (45) − (48) + (52) = (t − λ (\frac{n − 2}{2})) T^{2}(E_{n−1,n}(m ⊗ v)) = 2βT^{2}(X_{n−1}^{+}(m ⊗ v))

Using the lemma, we can now define the action of X_{0,1}^{±} and of H_{0,1} on F(M) by setting

X_{0,1}^{±}(m ⊗ v) = T^{−1}(X_{1,1}^{±}(T (m ⊗ v))) − βX_{0}^{±}(m ⊗ v)

and

H_{0,1}(m ⊗ v) = T^{−1}(H_{1,1}(T (m ⊗ v))) − βH_{0}(m ⊗ v).

Note that lemma 5.1 implies that X_{0,1}^{±}(T (m ⊗ v)) = T (X_{n−1,1}^{±}(m ⊗ v) + βX_{n−1}^{±}(m ⊗ v)) and similarly for H_{0,1}. In other words, and more generally, we set

φ(X_{0,r}^{±}) = T ∘ φ(ρ(X_{0,r}^{±})) ∘ T^{−1}, φ(H_{0,r}) = T ∘ φ(ρ(H_{0,r})) ∘ T^{−1} ∀r ≥ 0.

We now have to check that this indeed gives F(M) a structure of integrable module over LY. Choose i, j, k ∈ {0, 1, …, n − 1} with k ≠ i, k ≠ j. We have to verify that φ(X_{i,r}^{±}), φ(H_{i,r}), φ(X_{j,s}^{±}) and φ(H_{j,s}) satisfy the defining relations of LY. This is true when k = 0 from theorem 1 of [Dr1]. Using lemma 5.1, we conclude that it is also true for k ≠ 0. This means that we have a well-defined algebra homomorphism φ from LY to End_{ℂ}(F(M)). That F(M) is integrable follows from the fact that V^{⊗l} is an integrable 𝔰𝔩_{n}-module, and that it is of level l follows from theorem 4.1 in the case of ℂ[S_{l}] and 𝔘𝔰𝔩_{n}.

5.2 Proof of theorem 4.2, part 2

For the rest of this section, we assume that l + 2 < n. In the second step of the proof, we have to show that, given an integrable module M̂ of level l over LY, we can find a module M over H such that F(M) = M̂. Integrable 𝔘𝔰𝔩_{n}-modules are direct sums of finite dimensional ones, so, by the results of Drinfeld [Dr1] and Chari-Pressley [ChPr1], we know that there exists modules M^{1} and M^{2} over, respectively, H and ℂ[S̃_{l}], such that M̂ = F(M^{1}) as Y -module and M̂ = F(M^{2}) as L𝔰𝔩_{n}-module. Since ℂ[S_{l}] ⊂ H and ℂ[S_{l}] ⊂ ℂ[S̃_{l}], we have an isomorphism M^{1} ≅ M^{2} of S_{l}-modules, so we can denote them simply by M. We have to show that M is actually a module over H.

The following observation will be useful.

Lemma 5.2. If v = v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{l}} is a generator of V^{⊗l} as a module over 𝔘𝔰𝔩_{n} (that is, if i_{j} ≠ i_{k} for any j ≠ k), then m ⊗ v = 0 ⟹ m = 0.

Fix 1 ≤ j, k ≤ l, j ≠ k. We choose v to be the following generator of V^{⊗l} as 𝔘𝔰𝔩_{n}-module: v = v_{i_{1}} ⊗ v_{i_{2}} ⊗ ⋯ ⊗ v_{i_{l}} where i_{d} = d + 3 if d < j, d ≠ k, i_{d} = d + 2 if d > j, d ≠ k, i_{j} = 2 and i_{k} = 1.

We can express ω_{2}^{−} as an operator on V^{⊗l} in the following way:

ω_{2}^{−}|_{V^{⊗l}} = −\frac{1}{2} ∑_{d≠2,3^{d=1}}^{n} ∑_{r=1}^{l} ∑_{s≠r^{s=1}}^{l} sign(2 − d)(E_{3d}^{r}E_{d2}^{s}) − (\frac{n − 2}{4}) ∑_{r=1}^{l} E_{32}^{r} − \frac{1}{2} ∑_{r=1}^{l} ∑_{s≠r^{s=1}}^{l} E_{32}^{r}H_{2}^{s}

Therefore, [E_{n1}^{a}, ω_{2}^{−}] = −\frac{1}{2} ∑l ; r≠a^{r=1} E_{31}^{r}E_{n2}^{a} − \frac{1}{2} ∑l ; s≠a^{s=1} E_{31}^{a}E_{n2}^{s} and applying this to m ⊗ v with a = j, k

gives

[E_{n1}^{j}, ω_{2}^{−}](m ⊗ v) = −\frac{1}{2}E_{31}^{k}E_{n2}^{j}(m ⊗ v) and [E_{n1}^{k}, ω_{2}^{−}](m ⊗ v) = −\frac{1}{2}E_{31}^{k}E_{n2}^{j}(m ⊗ v).

(X_{2,1}^{−}X_{0}^{+} − X_{0}^{+}X_{2,1}^{−})(m ⊗ v) = ∑_{r=1}^{l} ∑_{s=1}^{l} (mx_{r}Y_{s} ⊗ X_{2,0}^{−,s}E_{n1}^{r}(v) − mY_{s}x_{r} ⊗ E_{n1}^{r}X_{2,0}^{−,s}(v))
−λ[ω_{2}^{−}, X_{0}^{+}](m ⊗ v)
= ∑_{r=1}^{l} ∑_{s=1}^{l} m[x_{r}, Y_{s}] ⊗ E_{32}^{s}E_{n1}^{r}(v) + λ ∑_{a=1}^{s} mx_{a} ⊗ [E_{n1}^{a}, ω_{2}^{−}](v)

(X_{2,1}^{−}X_{0}^{+} − X_{0}^{+}X_{2,1}^{−})(m ⊗ v) = m[x_{k}, Y_{j}] ⊗ E_{32}^{j}E_{n1}^{k}(v) − \frac{λ}{2}mx_{j} ⊗ E_{31}^{k}E_{n2}^{j}(v)

−\frac{λ}{2}mx_{k} ⊗ E_{31}^{k}E_{n2}^{j}(v)

= m([x_{k}, Y_{j}] − \frac{λ}{2}(x_{j} + x_{k})s_{jk}) ⊗ ṽ

where ṽ = E_{32}^{j}E_{n1}^{k}(v) = v_{a_{1}} ⊗ ⋯ ⊗ v_{a_{l}} with a_{d} = i_{d} if d ≠ j, k, a_{j} = 3 and a_{k} = n. We know from relation (3) that [X_{2,1}^{−}, X_{0}^{+}] = 0, so the last expression is equal to 0. Since ṽ is a generator of V^{⊗l} as a 𝔘𝔰𝔩_{n}-module, it follows, from lemma 5.2 and our assumption that λ = c, that m([x_{k}, Y_{j}] − \frac{c}{2}(x_{j} + x_{k})s_{jk}) = 0.

We consider now the relation between x_{k} and Y_{k}. From the definition of ν_{1}:

ν_{1} = \frac{1}{4} ∑_{d=3}^{n}(E_{1d}E_{d1} + E_{d1}E_{1d}) + \frac{1}{2}(E_{12}E_{21} + E_{21}E_{12}) − \frac{1}{4} ∑_{d=3}^{n}(E_{2d}E_{d2} + E_{d2}E_{2d}) − \frac{1}{2}H_{1}^{2}

whence, as an operator on V^{⊗l}, it is equal to

ν_{1}|_{V^{⊗l}} = \frac{1}{2} ∑_{d=3}^{n} ∑_{j=1}^{l} ∑_{s≠j^{s=1}}^{l}(E_{1d}^{j}E_{d1}^{s} − E_{2d}^{j}E_{d2}^{s}) + ∑_{j=1}^{l} ∑_{s≠j^{s=1}}^{l} E_{12}^{j}E_{21}^{s} − \frac{1}{2} ∑_{j=1}^{l} ∑_{s≠j^{s=1}}^{l} H_{1}^{j}H_{1}^{s} + (\frac{n − 2}{4}) ∑_{j=1}^{l} H_{1}^{j}.

Therefore,

[E_{n1}^{r}, ν_{1}] = \frac{1}{2} ∑_{d=3}^{n−1} ∑_{s≠r^{s=1}}^{l} E_{nd}^{r}E_{d1}^{s} + \frac{1}{2} ∑_{s≠r^{s=1}}^{l}(H_{0}^{r}E_{n1}^{s} + E_{21}^{r}E_{n2}^{s}) + ∑_{s≠r^{s=1}}^{l} E_{n2}^{r}E_{21}^{s} − ∑_{s≠r^{s=1}}^{l} E_{n1}^{r}H_{1}^{s} + (\frac{n − 2}{4}) E_{n1}^{r}.

Fix k, 1 ≤ k ≤ l. We now choose v to be equal to v = v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{l}} with i_{d} = d + 2 if d < k, i_{d} = d + 1 if d > k and i_{k} = 1. Note that v_{i_{d}} ≠ 2, n, n − 1 ∀d since l + 1 < n − 1 by assumption. Applying the previous expression for [E_{n1}^{r}, ν_{1}] to v, we obtain the following:

[E_{n1}^{r}, ν_{1}](v) = \frac{1}{2} ∑_{d=3}^{n} E_{nd}^{r}E_{d1}^{k}(v) = \frac{1}{2}s_{kr}E_{n1}^{k}(v) if r ≠ k [E_{n1}^{k}, ν_{1}](v) = (\frac{n − 2}{4}) E_{n1}^{k}(v). (60)

Note that H_{1,0}^{s}(v) = 0 if s ≠ k. We need (60) to obtain equation (61) below.

(H_{1,1}X_{0}^{+} − X_{0}^{+}H_{1,1})(m ⊗ v) = ∑_{r=1}^{l} ∑_{s=1}^{l} mx_{r}Y_{s} ⊗ H_{1,0}^{s}E_{n1}^{r}(v)

− ∑_{s=1}^{l} ∑_{r=1}^{l} mY_{s}x_{r} ⊗ E_{n1}^{r}H_{1,0}^{s}(v) − λ[ν_{1}, X_{0}^{+}](m ⊗ v)

= −mY_{k}x_{k} ⊗ E_{n1}^{k}H_{1,0}^{k}(v) + λ ∑_{r=1}^{l} mx_{r} ⊗ [E_{n1}^{r}, ν_{1}](v)

(H_{1,1}X_{0}^{+} − X_{0}^{+}H_{1,1})(m ⊗ v) = −mY_{k}x_{k} ⊗ E_{n1}^{k}(v) + \frac{λ}{2} ∑_{r≠k^{r=1}}^{l} mx_{r} ⊗ s_{kr}E_{n1}^{k}(v)

+λ (\frac{n − 2}{4}) m ⊗ E_{n1}^{k}(v)

= −mY_{k}x_{k} ⊗ ṽ + \frac{λ}{2} ∑_{r≠k^{r=1}}^{l} mx_{r}s_{kr} ⊗ ṽ + λ (\frac{n − 2}{4}) m ⊗ ṽ (61)

where ṽ = E_{n1}^{k}(v) = v_{a_{1}} ⊗ ⋯ ⊗ v_{a_{l}} with a_{d} = i_{d} if d ≠ k and a_{k} = n. We want to obtain a similar relation with H_{1,1} replaced by H_{n−1,1}.

From the definition of ν_{n−1},

ν_{n−1} = \frac{1}{4} ∑_{d=1}^{n−2}(E_{dn}E_{nd} + E_{nd}E_{dn}) + \frac{1}{2}(E_{n−1,n}E_{n,n−1} + E_{n,n−1}E_{n−1,n})

−\frac{1}{4} ∑_{d=1}^{n−2}(E_{d,n−1}E_{n−1,d} + E_{n−1,d}E_{d,n−1}) − \frac{1}{2}H_{n−1}^{2}

whence, as an operator on V^{⊗l}, it is equal to

ν_{n−1}|_{V^{⊗l}} = \frac{1}{2} ∑_{d=1}^{n−2} ∑_{j=1}^{l} ∑_{s≠j^{s=1}}^{l}(E_{dn}^{j}E_{nd}^{s} − E_{d,n−1}^{j}E_{n−1,d}^{s}) + ∑_{j=1}^{l} ∑_{s≠j^{s=1}}^{l}(E_{n−1,n}^{j}E_{n,n−1}^{s})

−\frac{1}{2} ∑_{j=1}^{l} ∑_{s≠j^{s=1}}^{l} H_{n−1}^{j}H_{n−1}^{s} − (\frac{n − 2}{4}) ∑_{j=1}^{l} H_{n−1}^{j}.

Therefore,

[E_{n1}^{r}, ν_{n−1}] = −\frac{1}{2} ∑_{d=2}^{n−2} ∑_{s≠r^{s=1}}^{l} E_{d1}^{r}E_{nd}^{s} + \frac{1}{2} ∑_{s≠r^{s=1}}^{l}(H_{0}^{r}E_{n1}^{s} − E_{n,n−1}^{r}E_{n−1,1}^{s})

− ∑_{s≠r^{s=1}}^{l} E_{n−1,1}^{r}E_{n,n−1}^{s} − ∑_{s≠r^{s=1}}^{l} E_{n1}^{r}H_{n−1}^{s} − (\frac{n − 2}{4}) E_{n1}^{r}.

Applying the previous expression for [E_{n1}^{r}, ν_{n−1}] to v, we conclude that [E_{n1}^{r}, ν_{n−1}](v) = 0 if r ≠ k and

[E_{n1}^{k}, ν_{n−1}](v) = −\frac{1}{2} ∑_{d=2}^{n−2} ∑_{s≠k^{s=1}}^{l} E_{d1}^{k}E_{nd}^{s}(v) − (\frac{n − 2}{4}) E_{n1}^{k}(v) = −\frac{1}{2} ∑_{s≠k^{s=1}}^{l} s_{ks}E_{n1}^{k}(v) − (\frac{n − 2}{4}) E_{n1}^{k}(v)

(62)

The equation (62) allows us to compute [H_{n−1,1}, X_{0}^{+}](m ⊗ v):

(H_{n−1,1}X_{0}^{+} − X_{0}^{+}H_{n−1,1})(m ⊗ v) = ∑_{r,s=1}^{l} (mx_{r}Y_{s} ⊗ H_{n−1,0}^{s}E_{n1}^{r}(v) − mY_{s}x_{r} ⊗ E_{n1}^{r}H_{n−1,0}^{s}(v))
−λ[ν_{n−1}, X_{0}](m ⊗ v)
= mx_{k}Y_{k} ⊗ H_{n−1,0}^{k}E_{n1}^{k}(v) + λ ∑_{r=1}^{s} mx_{r} ⊗ [E_{n1}^{r}, ν_{n−1}](v)

= −mx_{k}Y_{k} ⊗ E_{n1}^{k}(v) − \frac{λ}{2} ∑_{s≠k^{s=1}}^{l} mx_{k} (s_{ks} + \frac{n − 2}{2}) ⊗ E_{n1}^{k}(v)

= −mx_{k}Y_{k} ⊗ E_{n1}^{k}(v) − \frac{λ}{2} ∑_{s≠k^{s=1}}^{l} mx_{k} (s_{ks} + \frac{n − 2}{2}) ⊗ ṽ (63)

From the relations (1), (6) and (9) in LY, we know that

−X_{0,1}^{+} = [H_{1,1}, X_{0}^{+}] + ((λ − β)H_{1}X_{0}^{+} + βX_{0}^{+}H_{1})  (64)
= [H_{n−1,1}, X_{0}^{+}] + (βH_{n−1}X_{0}^{+} + (λ − β)X_{0}^{+}H_{n−1})  (65)

Applying these two expressions for −X_{0,1}^{+} to m ⊗ v, using equalities (61),(63) and the fact that H_{1}X_{0}^{+}(v) = 0 and X_{0}^{+}H_{n−1}(v) = 0 because of our choice of v, we obtain:

−mY_{k}x_{k} ⊗ ṽ + \frac{λ}{2} ∑_{r≠k^{r=1}}^{l} mx_{r} ⊗ s_{kr}ṽ + λ (\frac{n − 2}{4}) mx_{k} ⊗ ṽ + βX_{0}^{+}H_{1}(m ⊗ v) =

−mx_{k}Y_{k} ⊗ ṽ − \frac{λ}{2} ∑_{s≠k^{s=1}}^{l} mx_{k}s_{ks} ⊗ ṽ − λ (\frac{n − 2}{4}) mx_{k} ⊗ ṽ + βH_{n−1}X_{0}^{+}(m ⊗ v)

⟹ m[x_{k}, Y_{k}] ⊗ ṽ + \frac{λ}{2} ∑_{r≠k^{r=1}}^{l} m(x_{r} + x_{k})s_{kr} ⊗ ṽ + λ (\frac{n − 2}{2}) mx_{k} ⊗ ṽ + 2βmx_{k} ⊗ ṽ = 0

Since ṽ is a generator of V^{⊗l} as a 𝔘𝔰𝔩_{n}-module, it follows from lemma 5.2 and our assumptions that 2β + \frac{λ(n−2)}{2} = t, λ = c that

m([x_{k}, Y_{k}] + \frac{c}{2} ∑_{r≠k^{r=1}}^{l}(x_{r} + x_{k})s_{kr} + tx_{k}) = 0

We proved above that m([x_{k}, Y_{j}] − \frac{c}{2}(x_{j} + x_{k})s_{jk}) = 0 if j ≠ k. These last two equalities imply that M is a right module over H.

Therefore, we have shown that the H- and the ℂ[S̃_{l}]-module structure on M can be glued to yield a module over H. To prove that F is an equivalence, we are left to show that it is fully faithful. That F is injective on morphisms is true because this is true for the Schur-Weyl duality functor between ℂ[S̃_{l}] and 𝔘(L𝔰𝔩_{n}), so suppose that f : F(M_{1}) ⟶ F(M_{2}) is a LY -homomorphism. From the main results of [ChPr1] and [Dr1], f is of the form f(m_{1} ⊗ v) = g(m_{1}) ⊗ v, ∀m_{1} ∈ M_{1}, where g ∈ Hom_{ℂ}(M_{1}, M_{2}) is a linear map which is also a homomorphism of right ℂ[S̃_{l}]- and H-modules. Since H is generated by its two subalgebras ℂ[S̃_{l}] and H, g is even a homomorphism of H-modules. Therefore, f = F(g) and this completes the proof of theorem 4.2. □

6 Action of the elements X_{0,1}^{±}, H_{0,1}

Now that we know that F(M) is a module over LY, it may be interesting to see explicitly how the elements X_{0,1}^{±} and H_{0,1} act on it. What we will discover will be useful in the next section. We will assume throughout this section that λ = c and β = \frac{t}{2} − \frac{nc}{4} + \frac{c}{2}.

6.1 Action of X_{0,1}^{+}

Equations (64) and (65) yield

X_{0,1}^{+} = −\frac{1}{2}[H_{1,1} + H_{n−1,1}, X_{0}^{+}] − \frac{1}{2}(((λ − β)H_{1} + βH_{n−1})X_{0}^{+} + X_{0}^{+}(βH_{1} + (λ − β)H_{n−1})).

We will use the notation K_{r}(z) to denote the element z ⊗ u^{r} ∈ L𝔰𝔩_{n} for z ∈ 𝔰𝔩_{n}; in particular, K_{1}(E_{n1}) = X_{0}^{+} and K_{−1}(E_{1n}) = X_{0}^{−}. The element K_{r}(z) maps to the operator in End_{ℂ}(F(M)) given by K_{r}(z)(m ⊗ v) = ∑_{k=1}^{l} mx_{k}^{r} ⊗ z^{k}(v). Writing H_{1,1} as H_{1,1} = J(H_{1}) − λν_{1}, and similarly for H_{n−1,1}, we can express X_{0,1}^{+} in the following way. (We will use that [H_{n−1} − H_{1}, X_{0}^{+}] = 0.)

X_{0,1}^{+} = −\frac{1}{2}[J(H_{1} + H_{n−1}), X_{0}^{+}] − \frac{λ}{8} ∑_{d=3}^{n−1} (K_{1}(E_{nd})E_{d1} + E_{d1}K_{1}(E_{nd}))

+\frac{λ}{8}((K_{1}(E_{11}) − K_{1}(E_{nn}))E_{n1} + E_{n1}(K_{1}(E_{11}) − K_{1}(E_{nn}))) + \frac{λ}{4}(H_{1}X_{0}^{+} + X_{0}^{+}H_{1})

−\frac{λ}{4}(K_{1}(E_{n2})E_{21} + E_{21}K_{1}(E_{n2})) − \frac{λ}{8}(K_{1}(E_{21})E_{n2} + E_{n2}K_{1}(E_{21}))

+\frac{λ}{8} ∑_{d=2}^{n−2} (K_{1}(E_{d1})E_{nd} + E_{nd}K_{1}(E_{d1})) + \frac{λ}{8}((K_{1}(E_{11}) − K_{1}(E_{nn}))E_{n1}

+E_{n1}(K_{1}(E_{11}) − K_{1}(E_{nn}))) + \frac{λ}{4}(K_{1}(E_{n−1,1})E_{n,n−1} + E_{n,n−1}K_{1}(E_{n−1,1}))

+\frac{λ}{8}(K_{1}(E_{n,n−1})E_{n−1,1} + E_{n−1,1}K_{1}(E_{n,n−1}))

+\frac{λ}{4}(X_{0}^{+}H_{n−1} + H_{n−1}X_{0}^{+}) − \frac{1}{2}(((λ − β)H_{1} + βH_{n−1})X_{0}^{+} + X_{0}^{+}(βH_{1} + (λ − β)H_{n−1}))

X_{0,1}^{+} = −\frac{1}{2}[J(H_{1} + H_{n−1}), X_{0}^{+}] − \frac{λ}{8} ∑_{d=3}^{n−2} (K_{1}(E_{nd})E_{d1} + E_{d1}K_{1}(E_{nd}))

−\frac{λ}{4}(K_{1}(E_{n2})E_{21} + E_{21}K_{1}(E_{n2})) + \frac{λ}{8} ∑_{d=3}^{n−2} (K_{1}(E_{d1})E_{nd} + E_{nd}K_{1}(E_{d1}))

+\frac{λ}{4}((K_{1}(E_{11}) − K_{1}(E_{nn}))E_{n1} + E_{n1}(K_{1}(E_{11}) − K_{1}(E_{nn})))

+\frac{λ}{4}(K_{1}(E_{n−1,1})E_{n,n−1} + E_{n,n−1}K_{1}(E_{n−1,1}))

= −\frac{1}{2}[J(E_{11} − E_{nn}), X_{0}^{+}] + \frac{1}{2}[H_{2,1} + ⋯ H_{n−2,1}, X_{0}^{+}] + \frac{1}{2}[ν_{2} + ⋯ + ν_{n−2}, X_{0}^{+}]  (66)

−\frac{λ}{8} ∑_{d=2}^{n−2} (K_{1}(E_{nd})E_{d1} + E_{d1}K_{1}(E_{nd})) + \frac{λ}{8} ∑_{d=3}^{n−1} (K_{1}(E_{d1})E_{nd} + E_{nd}K_{1}(E_{d1}))

+\frac{λ}{8}(K_{1}(E_{n2})E_{21} + E_{21}K_{1}(E_{n2})) − \frac{λ}{4}(K_{1}(H_{0})E_{n1} + E_{n1}K_{1}(H_{0}))

+\frac{λ}{8}(K_{1}(E_{n−1,1})E_{n,n−1} + E_{n,n−1}K_{1}(E_{n−1,1}))

= \frac{1}{2}[J(E_{nn} − E_{11}), X_{0}^{+}] − \frac{λ}{8} ∑_{d=2}^{n−1} (K_{1}(E_{nd})E_{d1} + E_{d1}K_{1}(E_{nd}))

+\frac{λ}{8} ∑_{d=2}^{n−1} (K_{1}(E_{d1})E_{nd} + E_{nd}K_{1}(E_{d1})) − \frac{λ}{4}(K_{1}(H_{0})E_{n1} + E_{n1}K_{1}(H_{0}))

= J(X_{0}^{+}) − \frac{λ}{8} ∑_{ϵ∈Δ^{+}} ([X_{0}^{+}, X_{ϵ}^{+}]X_{ϵ}^{−} + X_{ϵ}^{−}[X_{0}^{+}, X_{ϵ}^{+}]) − \frac{λ}{8}(K_{1}(H_{0})E_{n1} + E_{n1}K_{1}(H_{0})).

We define J(X_{0}^{+}) to be equal to \frac{1}{2}[J(E_{nn} − E_{11}), X_{0}^{+}]. In line (66), we used the fact that

\frac{1}{2}[ν_{2} + ⋯ + ν_{n−2}, X_{0}^{+}] = \frac{1}{8}[E_{12}E_{21} + E_{21}E_{12} − E_{1,n−1}E_{n−1,1} + E_{2n}E_{n2} + E_{n2}E_{2n}

−E_{n−1,1}E_{1,n−1} − E_{n,n−1}E_{n−1,n} − E_{n−1,n}E_{n,n−1}, X_{0}^{+}]

Recall that H_{0} = E_{nn} − E_{11} = −H_{1} − … − H_{n−1}. Set Ỹ_{j} = \frac{1}{2}(x_{j}Y_{j} + Y_{j}x_{j}). The element J(X_{0}^{+}) acts on F(M) in the following way:

J(X_{0}^{+})(m ⊗ v) = \frac{1}{2} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m[x_{k}, Y_{j}] ⊗ H_{0}^{j}E_{n1}^{k}(v) + \frac{1}{2} ∑_{j=1}^{l} m(x_{j}Y_{j} + Y_{j}x_{j}) ⊗ E_{n1}^{j}(v)

= \frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(x_{k} + x_{j})s_{jk} ⊗ H_{0}^{j}E_{n1}^{k}(v) + ∑_{j=1}^{l} mỸ_{j} ⊗ E_{n1}^{j}(v)

J(X_{0}^{+})(m ⊗ v) = \frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(x_{k} + x_{j}) ⊗ (E_{nn}^{j}E_{n1}^{k} − E_{n1}^{j}E_{11}^{k})(v) + ∑_{j=1}^{l} mỸ_{j} ⊗ E_{n1}^{j}(v)

= (∑_{j=1}^{l} Ỹ_{j} ⊗ E_{n1}^{j} + \frac{λ}{8}(X_{0}^{+}H_{0} + H_{0}X_{0}^{+} + E_{n1}K_{1}(H_{0}) + K_{1}(H_{0})E_{n1})) (m ⊗ v)

Set J̃(X_{0}^{+}) = J(X_{0}^{+}) − \frac{λ}{8}(X_{0}^{+}H_{0} + H_{0}X_{0}^{+} + E_{n1}K_{1}(H_{0}) + K_{1}(H_{0})E_{n1}), so

X_{0,1}^{+} = J̃(X_{0}^{+}) − \frac{λ}{8} ∑_{ϵ∈Δ^{+}} ([X_{0}^{+}, X_{ϵ}^{+}]X_{ϵ}^{−} + X_{ϵ}^{−}[X_{0}^{+}, X_{ϵ}^{+}]) + \frac{λ}{8}(X_{0}^{+}H_{0} + H_{0}X_{0}^{+})

6.2 Action of H_{0,1}

We set ϵ_{0} = −ϵ_{1} − … − ϵ_{n−1} = −θ and ϵ^{0} = (ϵ, ϵ_{0}) for ϵ ∈ Δ^{+}. From relation (3), we know that [X_{0,1}^{+}, X_{0}^{−}] = H_{0,1}, so

H_{0,1} = [J̃(X_{0}^{+}), X_{0}^{−}] + \frac{λ}{4}H_{0}^{2} − \frac{λ}{4}(X_{0}^{+}X_{0}^{−} + X_{0}^{−}X_{0}^{+}) + \frac{λ}{4}(E_{1n}E_{n1} + E_{n1}E_{1n})

−\frac{λ}{4}K_{1}(H_{0})K_{−1}(H_{0}) − \frac{λ}{8} ∑_{ϵ≠−ϵ_{0}^{ϵ∈Δ^{+}}}(ϵ, ϵ_{0})(X_{ϵ}^{+}X_{ϵ}^{−} + X_{ϵ}^{−}X_{ϵ}^{+})

+\frac{λ}{8} ∑_{ϵ≠−ϵ_{0}^{ϵ∈Δ^{+}}}(ϵ, ϵ_{0})(K_{ϵ^{0}}(X_{ϵ}^{+})K_{−ϵ_{0}}(X_{ϵ}^{−}) + K_{−ϵ^{0}}(X_{ϵ}^{−})K_{ϵ^{0}}(X_{ϵ}^{+}))

= [J̃(X_{0}^{+}), X_{0}^{−}] − λν̃_{0}  (67)

where

ν̃_{0} = \frac{1}{8} ∑_{ϵ∈Δ^{+}}(ϵ, ϵ_{0})(X_{ϵ}^{+}X_{ϵ}^{−} + X_{ϵ}^{−}X_{ϵ}^{+} − K_{ϵ^{0}}(X_{ϵ}^{+})K_{−ϵ^{0}}(X_{ϵ}^{−}) − K_{−ϵ^{0}}(X_{ϵ}^{−})K_{ϵ^{0}}(X_{ϵ}^{+}))

−\frac{1}{4}H_{0}^{2} + \frac{1}{4}K_{1}(H_{0})K_{−1}(H_{0})

We can use this to find how H_{0,1} acts on F(M), so let us see explicitly how [J̃(X_{0}^{+}), X_{0}^{−}] acts on F(M):

[J̃(X_{0}^{+}), X_{0}^{−}](m ⊗ v) = ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m[x_{k}^{−1}, Ỹ_{j}] ⊗ E_{n1}^{j}E_{1n}^{k}(v)

+ ∑_{j=1}^{l} mx_{j}^{−1}Ỹ_{j} ⊗ E_{n1}^{j}E_{1n}^{j}(v) − ∑_{j=1}^{l} mỸ_{j}x_{j}^{−1} ⊗ E_{1n}^{j}E_{n1}^{j}(v)

[J̃(X_{0}^{+}), X_{0}^{−}](m ⊗ v) = −\frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(x_{j} + x_{k})(x_{j}^{−1} + x_{k}^{−1})s_{jk} ⊗ E_{n1}^{j}E_{1n}^{k}(v)

+\frac{1}{2} ∑_{j=1}^{l} m(Y_{j} + x_{j}^{−1}Y_{j}x_{j}) ⊗ E_{nn}^{j}(v) − \frac{1}{2} ∑_{j=1}^{l} m(x_{j}Y_{j}x_{j}^{−1} + Y_{j}) ⊗ E_{11}^{j}(v)

= −\frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(2 + x_{j}^{−1}x_{k} + x_{k}^{−1}x_{j}) ⊗ E_{11}^{j}E_{nn}^{k}(v) + ∑_{j=1}^{l} mY_{j} ⊗ H_{0}(v)

+\frac{1}{2} ∑_{j=1}^{l} m[x_{j}^{−1}, Y_{j}]x_{j} ⊗ E_{nn}^{j}(v) − \frac{1}{2} ∑_{j=1}^{l} mx_{j}[Y_{j}, x_{j}^{−1}] ⊗ E_{11}^{j}(v)

= −\frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(2 + x_{j}^{−1}x_{k} + x_{k}^{−1}x_{j}) ⊗ E_{11}^{j}E_{nn}^{k}(v) + ∑_{j=1}^{l} mY_{j} ⊗ H_{0}(v)

+\frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} mx_{k}(x_{j}^{−1} + x_{k}^{−1})s_{jk} ⊗ E_{nn}^{j}(v)

+\frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} mx_{j}(x_{j}^{−1} + x_{k}^{−1})s_{jk} ⊗ E_{11}^{j}(v) + \frac{t}{2}(E_{nn} + E_{11})(m ⊗ v)

= −λ(\frac{1}{2}E_{11}E_{nn} + \frac{1}{4}K_{−1}(E_{11})K_{1}(E_{nn}) + \frac{1}{4}K_{−1}(E_{nn})K_{1}(E_{11}))(m ⊗ v)

+ ∑_{j=1}^{l} mY_{j} ⊗ H_{0}(v) − \frac{λn}{4}E_{nn}(m ⊗ v) − \frac{λ}{2} ∑_{d=1}^{n} E_{dd}(m ⊗ v) − \frac{λn}{4}E_{11}(m ⊗ v)

+\frac{λ}{8} ∑_{d=1}^{n} (E_{nd}E_{dn} + E_{dn}E_{nd} + K_{1}(E_{nd})K_{−1}(E_{dn}) + K_{−1}(E_{dn})K_{1}(E_{nd}))(m ⊗ v)

+\frac{λ}{8} ∑_{d=1}^{n} (E_{1d}E_{d1} + E_{d1}E_{1d} + K_{−1}(E_{1d})K_{1}(E_{d1}) + K_{−1}(E_{d1})K_{−1}(E_{1d}))(m ⊗ v)

+\frac{t}{2}(E_{nn} + E_{11})(m ⊗ v)  (68)

Putting together equations (67) and (68), we conclude that H_{0,1} acts on F(M) in the following way:

H_{0,1}(m ⊗ v) = J(H_{0})(m ⊗ v) − \frac{λ}{2} ∑_{d=1}^{n} E_{dd}(m ⊗ v) + \frac{λ}{2}(E_{n1}E_{1n} + E_{1n}E_{n1})(m ⊗ v)

+\frac{λ}{4} ∑_{d=2}^{n−1}(E_{nd}E_{dn} + E_{dn}E_{nd} + E_{1d}E_{d1} + E_{d1}E_{1d})(m ⊗ v)

+\frac{λ}{2}H_{0}^{2}(m ⊗ v) + (β − \frac{λ}{2})(E_{11} + E_{nn})(m ⊗ v)

H_{0,1}(m ⊗ v) = J(H_{0})(m ⊗ v) − λν_{0}(m ⊗ v) − l (\frac{2t}{n} − λ) (m ⊗ v)

+(β − \frac{λ}{2}) ∑_{j=1}^{n}(\frac{E_{11} − E_{jj}}{n} + \frac{E_{nn} − E_{jj}}{n})(m ⊗ v)

where

ν_{0} = −\frac{1}{4} ∑_{d=2}^{n−1}(E_{nd}E_{dn} + E_{dn}E_{nd} + E_{1d}E_{d1} + E_{d1}E_{1d}) − \frac{1}{2}(E_{n1}E_{1n} + E_{1n}E_{n1}) − \frac{1}{2}H_{0}^{2}
= \frac{1}{4} ∑_{ϵ∈Δ^{+}}(ϵ, ϵ_{0})(X_{ϵ}^{+}X_{ϵ}^{−} + X_{ϵ}^{+}X_{ϵ}^{−}) − \frac{1}{2}H_{0}^{2}.

6.3 Action of X_{0,1}^{−}

Our goal now is to find elements that act on F(M) by m ⊗ v ↦ ∑_{j=1}^{l} my_{j} ⊗ E^{j}(v), E ∈ 𝔰𝔩_{n}. Set Y_{0}^{+} = \frac{1}{2}[X_{0}^{−}, H̃_{0,1}] where H̃_{0,1} = J(H_{0}) + λξ_{0} and

ξ_{0} = \frac{1}{4} ∑_{d=2}^{n−1}(E_{1d}E_{d1} + E_{d1}E_{1d} + E_{nd}E_{dn} + E_{dn}E_{nd}) + \frac{1}{2}(E_{n1}E_{1n} + E_{1n}E_{n1})

Y_{0}^{+}(m ⊗ v) = \frac{1}{2}[X_{0}^{−}, J(H_{0})](m ⊗ v) + \frac{λ}{2}[X_{0}^{−}, ξ_{0}](m ⊗ v) = \frac{1}{2} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m[Y_{j}, x_{k}^{−1}] ⊗ E_{1n}^{k}H_{0}^{j}(v)

+\frac{1}{2} ∑_{j=1}^{l} m(Y_{j}x_{j}^{−1} + x_{j}^{−1}Y_{j}) ⊗ E_{1n}^{j}(v) + \frac{λ}{2}[X_{0}^{−}, ξ_{0}](m ⊗ v)

= \frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(x_{j}^{−1} + x_{k}^{−1})s_{jk} ⊗ E_{1n}^{k}H_{0}^{j}(v) + \frac{1}{2} ∑_{j=1}^{l} my_{j} ⊗ E_{1n}^{j}(v)

+\frac{1}{4} ∑_{j=1}^{l} m(x_{j}y_{j}x_{j}^{−1} + x_{j}^{−1}y_{j}x_{j}) ⊗ E_{1n}^{j} + \frac{λ}{2}[X_{0}^{−}, ξ_{0}](m ⊗ v)

= \frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(x_{j}^{−1} + x_{k}^{−1})s_{jk} ⊗ E_{1n}^{k}H_{0}^{j}(v) + ∑_{j=1}^{l} my_{j} ⊗ E_{1n}^{j}(v)

+\frac{1}{4} ∑_{j=1}^{l} m(x_{j}[y_{j}, x_{j}^{−1}] + [x_{j}^{−1}, y_{j}]x_{j}) ⊗ E_{1n}^{j}(v) + \frac{λ}{2}[X_{0}^{−}, ξ_{0}](m ⊗ v)

= \frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(x_{j}^{−1} + x_{k}^{−1})s_{jk} ⊗ E_{1n}^{k}H_{0}^{j}(v) + ∑_{j=1}^{l} my_{j} ⊗ E_{1n}^{j}(v) + \frac{λ}{2}[X_{0}^{−}, ξ_{0}](m ⊗ v)

+\frac{1}{4} ∑_{j=1}^{l} m(x_{j}( − tx_{j}^{−2} − λ ∑_{k≠j} x_{j}^{−1}x_{k}^{−1}s_{jk}) + (tx_{j}^{−2} + λ ∑_{k≠j} x_{j}^{−1}x_{k}^{−1}s_{jk})x_{j}) ⊗ E_{1n}^{j}(v)

= \frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(x_{j}^{−1} + x_{k}^{−1})s_{jk} ⊗ E_{1n}^{k}H_{0}^{j}(v) + ∑_{j=1}^{l} my_{j} ⊗ E_{1n}^{j}(v)

+\frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(x_{j}^{−1} − x_{k}^{−1})s_{jk} ⊗ E_{1n}^{j}(v) + \frac{λ}{2}[X_{0}^{−}, ξ_{0}](m ⊗ v)

= \frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} m(x_{j}^{−1} + x_{k}^{−1}) ⊗ (E_{nn}^{k}E_{1n}^{j} − E_{1n}^{k}E_{11}^{j})(v)

+ ∑_{j=1}^{l} my_{j} ⊗ E_{1n}^{j}(v) + \frac{λ}{2}[X_{0}^{−1}, ξ_{0}](m ⊗ v) + \frac{λ}{4} ∑_{j=1}^{l} ∑_{k≠j^{k=1}}^{l} ∑_{d=1}^{n} m(x_{j}^{−1} − x_{k}^{−1}) ⊗ E_{1d}^{k}E_{dn}^{j}

= \frac{λ}{8}(E_{nn}K_{−1}(E_{1n}) + K_{−1}(E_{1n})E_{nn} + K_{−1}(E_{nn})E_{1n} + E_{1n}K_{−1}(E_{nn}))(m ⊗ v)

−\frac{λ}{8}(E_{1n}K_{−1}(E_{11}) + K_{−1}(E_{11})E_{1n} + K_{−1}(E_{1n})E_{11} + E_{11}K_{−1}(E_{1n}))(m ⊗ v)

+ ∑_{j=1}^{l} my_{j} ⊗ E_{1n}^{j}(v) + \frac{λ}{2}[X_{0}^{−}, ξ_{0}]

+\frac{λ}{8} ∑_{d=1}^{n} (E_{1d}K_{−1}(E_{dn}) + K_{−1}(E_{dn})E_{1d})(m ⊗ v) − \frac{nλ}{8}K_{−1}(E_{1n})(m ⊗ v)

−\frac{λ}{8} ∑_{d=1}^{n} (K_{−1}(E_{1d})E_{dn} + E_{dn}K_{−1}(E_{1d}))(m ⊗ v) + \frac{nλ}{8}K_{−1}(E_{1n})(m ⊗ v)

= ∑_{j=1}^{l} my_{j} ⊗ E_{1n}^{j}(v)

It is actually easy to express Y_{0}^{+} in terms of the generators of LY given in definition 3.2. Since H_{0} = −H_{1} − … − H_{n−1}, we can write

H̃_{0,1} = J(H_{0}) + λξ_{0} = −J(H_{1}) − … − J(H_{n−1}) + λξ_{0}
= −(H_{1,1} + … H_{n−1,1}) − λ(ν_{1} + … + ν_{n−1}) + λξ_{0}
= −(H_{1,1} + … H_{n−1,1}) + \frac{λ}{2}(H_{1}^{2} + … + H_{n−1}^{2})

Therefore, using relations (6) and (9), we obtain

Y_{0}^{+} = \frac{1}{2}[X_{0}^{−}, H̃_{0,1}]
= −\frac{1}{2}[X_{0}^{−}, H_{1,1}] − \frac{1}{2}[X_{0}^{−}, H_{n−1,1}] − \frac{λ}{4}(X_{0}^{−}H_{1} + H_{1}X_{0}^{−}) − \frac{λ}{4}(X_{0}^{−}H_{n−1} + H_{n−1}X_{0}^{−})
= \frac{1}{2}X_{0,1}^{−} + \frac{1}{2}(βH_{1}X_{0}^{−} + (λ − β)X_{0}^{−}H_{1}) + \frac{1}{2}X_{0,1}^{−} + \frac{1}{2}((λ − β)H_{n−1}X_{0}^{−} + βX_{0}^{−}H_{n−1})

−\frac{λ}{4}(X_{0}^{−}H_{1} + H_{1}X_{0}^{−}) − \frac{λ}{4}(X_{0}^{−}H_{n−1} + H_{n−1}X_{0}^{−})

= X_{0,1}^{−}

The element X_{0,1}^{−} will become important in the next section.

It can be seen from the relations in LY that the subalgebra of LY generated by Y_{0}^{+} and 𝔰𝔩_{n} is isomorphic to 𝔘(𝔰𝔩_{n} ⊗_{ℂ} ℂ[u]). (See also proposition 7.1 below.) We introduce the notation Q_{r}(z), r ∈ ℤ_{≥0}, to denote z ⊗ u^{r} as an element of this subalgebra; in particular, Q_{1}(E_{1n}) = Y_{0}^{+}. There are three types of operators in End_{ℂ}(M ⊗_{ℂ[S_{l}]} V^{⊗l}) which are of particular interest to us: those coming from the action of J(z), K_{r}(z) and Q_{r}(z). They are related to each other in the following way.

Proposition 6.1 (See also [BHW]). Suppose that a ≠ b and c ≠ d. Then we have the equality [Q_{1}(E_{ab}), K_{1}(E_{cd})] + [K_{1}(E_{ab}), Q_{1}(E_{cd})] = 2(δ_{bc}J(E_{ad}) − δ_{da}J(E_{cb})).

Proof. First, we will prove the equality

[Q_{1}(E_{1n}), K_{1}(H_{0})] + [K_{1}(E_{1n}), Q_{1}(H_{0})] = 4J(E_{1n})  (69)

[Q_{1}(E_{1n}), K_{1}(H_{0})] + [K_{1}(E_{1n}), Q_{1}(H_{0})] = [X_{0,1}^{−}, [X_{0}^{+}, E_{1n}]] + \frac{1}{2}[[E_{1n}, [X_{0}^{+}, E_{1n}]], [E_{n1}, X_{0,1}^{−}]]
= −[H_{0,1}, E_{1n}] − \frac{1}{2}[[H_{0}, [X_{0}^{+}, E_{1n}]], X_{0,1}^{−}]

−\frac{1}{2}[[E_{1n}, [X_{0}^{+}, H_{0}]], X_{0,1}^{−}]

+\frac{1}{2}[E_{n1}, [E_{1n}, [[X_{0}^{+}, X_{0,1}^{−}], E_{1n}]]]

= −[H_{0,1}, E_{1n}] + [[E_{1n}, X_{0}^{+}], X_{0,1}^{−}]

+\frac{1}{2}[E_{n1}, [E_{1n}, [H_{0,1}, E_{1n}]]]

= 2[E_{1n}, H_{0,1}] + \frac{1}{2}[E_{n1}, [E_{1n}, [H_{0,1}, E_{1n}]]]  (70)

[H_{0,1}, E_{1n}] = [[H_{0,1}, E_{12}], E_{2n}] + [E_{12}, [E_{23}, [⋯ [E_{n−2,n−1}, [H_{0,1}, E_{n−1,n}] ⋯]
= [−X_{1,1}^{+} − (βH_{0}X_{1}^{+} + (λ − β)X_{1}^{+}H_{0}), E_{2n}]

+[E_{12}, [E_{23}, [⋯ [E_{n−2,n−1}, −X_{n−1,1}^{+} − ((λ − β)H_{0}X_{n−1}^{+} + βX_{n−1}^{+}H_{0})] ⋯]

[H_{0,1}, E_{1n}] = [−J(X_{1}^{+}) + λω_{1}^{+} − (βH_{0}X_{1}^{+} + (λ − β)X_{1}^{+}H_{0}), E_{2n}] +

[E_{12}, [E_{23}, [⋯ [E_{n−2,n−1}, −J(X_{n−1}^{+}) + λω_{n−1}^{+} − ((λ − β)H_{0}X_{n−1}^{+} + βX_{n−1}^{+}H_{0})] · ]

[H_{0,1}, E_{1n}] = −J(E_{1n}) − (βH_{0}E_{1n} + (λ − β)E_{1n}H_{0}) + (βE_{2n}E_{12} + (λ − β)E_{12}E_{2n})
+λ[ω_{1}^{+}, E_{2n}] − J(E_{1n}) + λ[E_{1,n−1}, ω_{n−1}^{+}]
−[E_{12}, [E_{23}, [⋯ [E_{n−2,n−1}, ((λ − β)H_{0}X_{n−1}^{+} + βX_{n−1}^{+}H_{0})] ⋯]  (71)

The expression [E_{12}, [E_{23}, [⋯ [E_{n−2,n−1}, ((λ − β)H_{0}X_{n−1}^{+} + βX_{n−1}^{+}H_{0})] ⋯] is equal to

[E_{12}, (λ − β)H_{0}E_{2n} + βE_{2n}H_{0}] = ((λ − β)E_{12}E_{2n} + βE_{2n}E_{12}) + ((λ − β)H_{0}E_{1n} + βE_{1n}H_{0}) (72)

[E_{1,n−1}, ω_{n−1}^{+}] = −\frac{1}{4} ∑_{j=2}^{n−2}(E_{jn}E_{1j} + E_{1j}E_{jn}) − \frac{1}{4}(E_{1n}(E_{11} − E_{n−1,n−1}) + (E_{11} − E_{n−1,n−1})E_{1n})

−\frac{1}{4}(E_{1n}H_{n−1} + H_{n−1}E_{1n}) − \frac{1}{4}(E_{n−1,n}E_{1,n−1} + E_{1,n−1}E_{n−1,n})

= −\frac{1}{4} ∑_{j=2}^{n−1}(E_{jn}E_{1j} + E_{1j}E_{jn}) + \frac{1}{4}(E_{1n}H_{0} + H_{0}E_{1n})  (73)

[ω_{1}^{+}, E_{2n}] = \frac{1}{4} ∑_{j=3}^{n−1}(E_{jn}E_{1j} + E_{1j}E_{jn}) − \frac{1}{4}((E_{22} − E_{nn})E_{1n} + E_{1n}(E_{22} − E_{nn}))

−\frac{1}{4}(E_{1n}H_{1} + H_{1}E_{1n}) + \frac{1}{4}(E_{12}E_{2n} + E_{2n}E_{12})

= \frac{1}{4} ∑_{j=2}^{n−1}(E_{jn}E_{1j} + E_{1j}E_{jn}) + \frac{1}{4}(E_{1n}H_{0} + H_{0}E_{1n})  (74)

Therefore, combining equations (71),(73),(74) and (72), we obtain the following simple expression for [H_{0,1}, E_{1n}]:

[H_{0,1}, E_{1n}] = −2J(E_{1n}) − \frac{λ}{2}(H_{0}E_{1n} + E_{1n}H_{0})  (75)

Putting together equations (70) and (75) yields equality (69):

[Q(E_{1n}), K_{1}(H_{0})] + [K_{1}(E_{1n}), Q(H_{0})] = 4J(E_{1n}) + λ(H_{0}E_{1n} + E_{1n}H_{0})

−\frac{1}{2}[E_{n1}, [E_{1n}, 2J(E_{1n}) + \frac{λ}{2}(H_{0}E_{1n} + E_{1n}H_{0})]]

= 4J(E_{1n}) + λ(H_{0}E_{1n} + E_{1n}H_{0}) − λ[E_{n1}, E_{1n}^{2}]
= 4J(E_{1n})

Applying [E_{n1}, ·] to both sides of equation (69) yields

[K_{1}(E_{n1}), Q_{1}(E_{1n})] + [Q_{1}(E_{n1}), K_{1}(E_{1n})] = 2J(H_{0})  (76)

This proves proposition 6.1 when a = n, b = 1, c = 1, d = n. If a = n, b = c = 1, d ≠ 1, n, we apply [·, E_{nd}] to the previous equation to obtain [K_{1}(E_{n1}), Q_{1}(E_{1d})] + [Q_{1}(E_{n1}), K_{1}(E_{1d})] = 2J(E_{nd}). To obtain equation (6.1) for a = n, b = 1, c ≠ 1, d, d ≠ n, 1, we use [E_{c1}, ·], which yields [K_{1}(E_{n1}), Q_{1}(E_{cd})] + [Q_{1}(E_{n1}), K_{1}(E_{cd})] = 0. Under the assumption c ≠ 1, we apply [·, E_{d1}] to the previous equation and get [K_{1}(E_{n1}), Q_{1}(E_{c1})] + [Q_{1}(E_{n1}), K_{1}(E_{c1})] = 0. Apply [E_{c1}, ·] if c ≠ 1, n to (76) give [K_{1}(E_{n1}), Q_{1}(E_{cn})] + [Q_{1}(E_{n1}), K_{1}(E_{cn})] = −2J(E_{c1}). We have covered all the cases with a = n, b = 1.

Now suppose that a = n but b̃ ≠ 1. Equation (6.1) in the case a = n, b = c = 1, , b ≠ n, d ≠ 1 along with [·, E_{1b̃}] yields [K_{1}(E_{nb̃}), Q_{1}(E_{1d})] + [Q_{1}(E_{nb̃}), K_{1}(E_{1d})] = −2δ_{ad}J(E_{1b̃}). To recover the case when c ≠ 1, b̃, we apply [E_{c1}, ·] to the previous equation, so we have shown the equality

[K_{1}(E_{nb̃}), Q_{1}(E_{cd})] + [Q_{1}(E_{nb̃}), K_{1}(E_{cd})] = −2δ_{ad}J(E_{cb̃}).  (77)

Using equation (77) along with [E_{b̃c}, ·] yields equation (6.1) in the case a = n and b = c, d are arbitrary. The remaining cases can be obtained using similar computations.

7 Schur-Weyl dual of the rational Cherednik algebra

Our goal in this section is to establish an equivalence of categories for the rational Cherednik algebra similar to the one given in theorem 4.2 and to identify the Schur-Weyl dual of H with a subalgebra of LY.

7.1 Case of type 𝔤𝔩_{l}

Definition 7.1. The subalgebra of LY generated by X_{i}^{±}, 1 ≤ i ≤ n − 1, X_{0}^{+} and Y_{0}^{+} is denoted by 𝕃_{β,λ} and called a Yangian deformed double-loop algebra, as suggested in [BHW]. The polynomial loop algebra generated by X_{i}^{±}, 1 ≤ i ≤ n − 1 and X_{0}^{+} (resp. Y_{0}^{+}) is denoted L_{X} (resp. L_{Y}).

Remark 7.1. The algebra LY_{β,λ} is the same as the subalgebra generated by z, K_{1}(z), Q_{1}(z), ∀z ∈ 𝔰𝔩_{n}. Furthermore, proposition 6.1 implies that 𝕃_{β,λ} contains all the elements X_{i,r}^{±}, H_{i,r} for 1 ≤ i ≤ n, r ≥ 0 and relation (12) shows that it also contains X_{0,r}^{+}, ∀r ≥ 0 and X_{0,r}^{−}, ∀r ≥ 1. We will abbreviate 𝕃_{β,λ} by 𝕃.

The computations for the action of X_{0,1}^{−} on M ⊗_{ℂ[S_{l}]} V^{l} and the anti-symmetric role of 𝔥 and 𝔥^{∗} in the definition of H_{t,c}, along with the last proposition of the previous section, suggest that the following result is true.

Proposition 7.1. There exists an anti-involution ι of 𝕃 which interchanges L_{X} and L_{Y} and which is given on the generators by the formulas

ι(X_{i,r}^{±}) = X_{i,r}^{∓} if i ≠ 0, ι(H_{i,r}) = H_{i,r}

ι(X_{0,r}^{+}) = X_{0,r+1}^{−} for r ≥ 0, ι(X_{0,r}^{−}) = X_{0,r−1}^{+} for r ≥ 1

Proof. This can be checked using the relations given in definition 3.2.

Theorem 7.1. Suppose that l ≥ 1, n ≥ 3. Set λ = c and β = \frac{t}{2} − \frac{cn}{4} + \frac{c}{2}. The functor M ↦ M ⊗_{ℂ[S_{l}]} V^{⊗l} sends a right H-module to an integrable left 𝕃-module of level l. Furthermore, if l + 2 < n, this functor is an equivalence.

Proof. As for theorem 4.2, the proof is in two parts. First, it is enough to take M = H and show that F(M) is a module over 𝕃. We can view H ⊗_{ℂ[S_{l}]}ℂ^{⊗l} as a subspace of H ⊗_{ℂ[S_{l}]}ℂ^{⊗l}; the later is a module over 𝕃 since it is even a module over LY. The subspace F(H) is stable under the action of the subalgebras L_{X} and L_{Y}, so it is a module over the subalgebra of LY generated by L_{X} and L_{Y}, which is exactly 𝕃. The fact that F(M) is integrable of level l follows from the same argument as in the proof of theorem 4.2.

Now let N be an integrable module of level l over 𝕃 and suppose that l+2 < n. We have to show that there exists a module M over H such that F(M) = N. We can argue as for the trigonometric case to conclude that there exists an S_{l}-module M, which is also a ℂ[𝔥] ⋊ W - and a ℂ[𝔥^{∗}] ⋊ W -module, such that F(M) ≅ N. As before, we must show that M is actually a module over H.

Fix 1 ≤ j, k ≤ l, j ≠ k. Choose v = v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{l}} such that i_{k} = 2, i_{j} = n − 1, i_{r} = r + 2 if r < j, r ≠ k, i_{r} = r + 1 if r > j, r ≠ k. Set ṽ = E_{n2}^{k}E_{1,n−1}^{j}(v).

On one hand,

(Q_{1}(E_{1,n−1})K_{1}(E_{n2}) − K_{1}(E_{n2})Q_{1}(E_{1,n−1}))(m ⊗ v) =
∑_{s=1}^{l} ∑_{r=1}^{l} mx_{r}y_{s} ⊗ E_{1,n−1}^{s}E_{n2}^{r}(v) − ∑_{s=1}^{l} ∑_{r=1}^{l} my_{s}x_{r} ⊗ E_{n2}^{r}E_{1,n−1}^{s}(v) = m(x_{k}y_{j} − y_{j}x_{k}) ⊗ ṽ (78)

On the other hand, Q_{1}(E_{1,n−1}) = [Y_{0}^{+}, E_{n,n−1}] and K_{1}(E_{n2}) = [X_{0}^{+}, E_{12}], so:

[Q_{1}(E_{1,n−1}), K_{1}(E_{n2})] = [[Y_{0}^{+}, E_{n,n−1}], [X_{0}^{+}, E_{12}]] = [[X_{0,1}^{−}, [X_{0}^{+}, E_{12}]], E_{n,n−1}]
= [[[X_{0,1}^{−}, X_{0}^{+}], X_{1}^{+}], E_{n,n−1}] = −[[H_{0,1}, X_{1}^{+}], E_{n,n−1}]
= −[−X_{1,1}^{+} − (βH_{0}X_{1}^{+} + (λ − β)X_{1}^{+}H_{0}), X_{n−1}^{−}]
= [βH_{0}X_{1}^{+} + (λ − β)X_{1}^{+}H_{0}, X_{n−1}^{−}]
= βE_{n,n−1}E_{12} + (λ − β)E_{12}E_{n,n−1}

Therefore,

[Q_{1}(E_{1,n−1}), K_{1}(E_{n2})](m ⊗ v) = m ⊗ (βE_{n,n−1}E_{12} + (λ − β)E_{12}E_{n,n−1})(v)
= λm ⊗ E_{12}^{k}E_{n,n−1}^{j}(v) = λms_{jk} ⊗ ṽ  (79)

Equations (78) and (79) imply that m(x_{k}y_{j} − y_{j}x_{k} − λs_{jk}) ⊗ ṽ = 0. From lemma 5.2 and our assumption that λ = c, we conclude that

m(x_{k}y_{j} − y_{j}x_{k} − cs_{jk}) = 0.  (80)

Now let v be determined by i_{k} = n − 1, i_{j} = j + 1 if j ≠ k. Set v̂ = E_{n,n−1}^{k}(v). On one hand,

[K_{1}(E_{n1}), Q_{1}(E_{1,n−1})](m ⊗ v) = my_{k}x_{k} ⊗ E_{n1}^{k}E_{1,n−1}^{k}(v) = my_{k}x_{k} ⊗ v̂  (81)

On the other hand,

[K_{1}(E_{n1}), Q_{1}(E_{1,n−1})] = [X_{0}^{+}, [Y_{0}^{−}, E_{n,n−1}]] = [X_{0}^{+}, [X_{0,1}^{−}, X_{n−1}^{−}]]
= [H_{0,1}, X_{n−1}^{−}] = X_{n−1,1}^{−} + (βH_{0}X_{n−1}^{−} + (λ − β)X_{n−1}^{−}H_{0})
= J(X_{n−1}^{−}) − λω_{n−1}^{−} + (βH_{0}X_{n−1}^{−} + (λ − β)X_{n−1}^{−}H_{0})

where

ω_{n−1}^{−} = −\frac{1}{4} ∑_{d=1}^{n−2}(E_{nd}E_{d,n−1} + E_{d,n−1}E_{nd}) − \frac{1}{4}(X_{n−1}^{−}H_{n−1} + H_{n−1}X_{n−1}^{−}).

Therefore, we also have:

[K_{1}(E_{n1}), Q_{1}(E_{1,n−1})](m ⊗ v) = (J(X_{n−1}^{−}) − λω_{n−1}^{−} + (βH_{0}X_{n−1}^{−} + (λ − β)X_{n−1}^{−}H_{0}))(m ⊗ v)
= mY_{k} ⊗ E_{n,n−1}^{k}(v) − λm ⊗ (ω_{n−1}^{−}(v)) + βm ⊗ H_{0}^{k}E_{n,n−1}^{k}(v)
= mY_{k} ⊗ v̂ + \frac{λ}{2} ∑_{d=1}^{n−2} ∑_{r=1}^{l} ∑_{s≠d^{s=1}}^{l} m ⊗ (E_{nd}^{s}E_{d,n−1}^{r})(v)

+λ (\frac{n − 2}{4}) m ⊗ E_{n,n−1}(v) + \frac{λ}{4}m ⊗ (X_{n−1}^{−}H_{n−1}

+H_{n−1}X_{n−1}^{−})(v) + βm ⊗ H_{0}^{k}E_{n,n−1}^{k}(v)

= mY_{k} ⊗ v̂ + \frac{λ}{2} ∑_{d=1}^{n−2} ∑_{j≠k^{j=1}}^{l} m ⊗ (E_{nd}^{j}E_{d,n−1}^{k})(v) + λ (\frac{n − 2}{4}) m ⊗ v̂

+\frac{λ}{4}m(E_{n,n−1}^{k}E_{n−1,n−1}^{k} − E_{nn}^{k}E_{n,n−1}^{k})(v) + βm ⊗ E_{n,n−1}^{k}(v)

= \frac{1}{2}m(x_{k}y_{k} + y_{k}x_{k}) ⊗ v̂ + \frac{λ}{2} ∑_{j≠k^{j=1}}^{l} ms_{jk} ⊗ v̂

+(\frac{λn}{4} − \frac{λ}{2} + β)m ⊗ v̂  (82)

From the equations (81) and (82) and our hypothesis that β = \frac{t}{2} − \frac{λn}{4} + \frac{λ}{2}, we deduce the following equality:

my_{k}x_{k} ⊗ v̂ = \frac{1}{2}m(x_{k}y_{k} + y_{k}x_{k}) ⊗ v̂ + \frac{λ}{2} ∑_{j≠k^{j=1}}^{l} ms_{jk} ⊗ v̂ + \frac{t}{2}m ⊗ v̂

which implies that m(y_{k}x_{k} − x_{k}y_{k} − t − λ ∑l ; j≠k^{j=1} s_{jk}) ⊗ v̂ = 0. Since λ = c by assumption and v̂ is

a generator of V^{⊗l} as 𝔘𝔰𝔩_{n}-module, we conclude, using again lemma 5.2, that the equality

m(y_{k}x_{k} − x_{k}y_{k} − t − c ∑_{j≠k^{j=1}}^{l} s_{jk}) = 0  (83)

must be satisfied. Equations (80) and (83) show that M is a right module over H. Finally, that F is bijective on the set of morphisms follows from an argument similar to the one used in the trigonometric case.

7.2 Category O

One important category of modules over H_{t,c} (when t ≠ 0) is the category O studied in [GGOR].

Definition 7.2. We define O_{t,c} for t ≠ 0 to be the category of right modules over H_{t,c} which are finitely generated over H_{t,c} and locally nilpotent over ℂ[𝔥^{∗}]. We set O = O_{t,c}.

We see from the definition of the 𝕃-module structure on F(M) that if M ∈ O then F(M) is locally nilpotent over the subalgebra A of 𝕃 generated by Q_{r}(z), ∀z ∈ 𝔰𝔩_{n}, ∀r ≥ 1. This leads us to our last theorem.

Theorem 7.2. Assume that l + 2 < n, λ = c and β = \frac{t}{2} − \frac{cn}{4} + \frac{c}{2}. The functor F establishes an equivalence between the category O and the category of finitely generated left modules over 𝕃 which are locally nilpotent over the subalgebra A and integrable of level l.

Proof. We prove this theorem for H. If m_{1}, …, m_{k} are generators of M, then {m_{i} ⊗ v, 1 ≤ i ≤ k, v ∈ V^{l}} is a finite set of generators for F(M). To see this, we can assume that M is generated over ℂ[𝔥] by m_{1}, …, m_{k}. Take an element m ⊗ v ∈ F(M) with m = m_{1}x_{1}^{a_{1}} ⋯ x_{l}^{a_{l}}. We suppose first that v = v_{1} ⊗ v_{2} ⊗ v_{3} ⊗ ⋯ ⊗ v_{l} and set v′ = v_{1} ⊗ v_{3} ⊗ v_{4} ⊗ ⋯ ⊗ v_{l+1}. Then

m ⊗ v = K_{a_{l}}(E_{l,l+1}) ⋯ K_{a_{2}}(E_{23})K_{a_{1}}(H_{1})(m_{1} ⊗ v′).

Now we can apply elements of 𝔘𝔰𝔩_{n} to v_{1} ⊗ v_{2} ⊗ ⋯ ⊗ v_{l} to obtain any other element of V^{l}. The general case when m = ∑_{j=1}^{k} m_{j}x_{1}^{a_{1,j}} ⋯ x_{l}^{a_{l,j}} ⊗ v_{j} follows from this. Conversely, suppose that N is a finitely generated integrable module over 𝕃 of level l and N = F(M). Let {n_{1}, …, n_{k}} be a set of generators of N and write n_{i} = ∑_{j=1}^{k_{i}} m_{ij} ⊗ v_{ij} for some m_{ij} ∈ M and some v_{ij} ∈ V^{l}. Then {m_{ij}|1 ≤ i ≤ k, 1 ≤ j ≤ k_{i}} is a set of generators of M.

Now suppose that N is an integrable left module over 𝕃 of level l which is locally nilpotent over A. By theorem 7.1, we know that N = F(M) for a right module M over H. Pick m ∈ M. It is enough to show that my_{i}^{s} = 0 for some s. Set v = v_{1} ⊗ ⋯ ⊗ v_{i} ⊗ v_{i+2} ⊗ ⋯ ⊗ v_{l+1} and choose s so that Q_{1}(H_{i})^{s}(m ⊗ v) = 0. Then Q_{1}(H_{i})(m ⊗ v) = my_{i} ⊗ v, so my_{i}^{s} ⊗ v = Q_{1}(H_{i})^{s}(m ⊗ v) = 0 and lemma 5.2 implies that my_{i}^{s} = 0.

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Korteweg-De Vries Institute for Mathematics University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands E-mail: nguay@science.uva.nl

and

Department of Mathematics University of Illinois at Urbana-Champaign Altgeld Hall 1409 W. Green Street Urbana, IL 61801 USA E-mail: nguay@math.uiuc.edu