Cherednik algebras and Yangians Nicolas Guay

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1 Cherednik agebras and Yangians Nicoas Guay Abstract We construct a functor from the category of modues over the trigonometric resp. rationa Cherednik agebra of type g to the category of integrabe modues of eve over a Yangian for the oop agebra s n resp. over a subagebra of this oop Yangian and we estabish that it is an equivaence of categories if < n. Finay, we treat the case of the rationa Cherednik agebras of type A 1. 1 Introduction Affine Hecke agebras are very important in representation theory and have been studied extensivey over the past few decades, aong with their degenerate version introduced in [Dr1] and in [Lu]. About fifteen years ago, I. Cherednik introduced the notion of doube affine Hecke agebra [Ch], abbreviated DAHA, which he used to prove some important conectures of I. Macdonad. His agebra aso admits degenerate versions, the trigonometric one and the rationa one, which are caed Cherednik agebras. One of the most important cassica resuts in representation theory is an equivaence, often caed Schur-Wey duaity, between the category of modues over the symmetric group S and the category of modues of eve over the Lie agebra s n for n 1. When quantum groups were invented in the 1980 s, it became an interesting probem to generaize the Schur-Wey correspondence and simiar equivaences were obtained between finite Hecke agebras and quantized enveoping agebras [Ji], between degenerate affine Hecke agebras and Yangians [Dr1, Ch1], between affine Hecke agebras and quantized affine Lie agebras [GRV, ChPr1], and between doube affine Hecke agebras and toroida quantum agebras [VaVa1]. In this paper, we prove a simiar equivaence of categories between the trigonometric resp. rationa Cherednik agebra associated to the symmetric group S and a resp. subagebra L of a Yangian LY for the oop agebra Ls n = s n C C[u, u 1 ]. The oop Yangians are barey known. They were mentioned briefy in [Va], [VaVa].See aso [BoLe] for the ŝ case. The sub-agebra on the other side of our equivaence from the rationa Cherednik agebra has never been considered before. By contrast, there has been a recent surge of interest in the representation theory of Cherednik agebras and their reations to the geometry of Hibert schemes, integrabe systems and other important mathematica obects. See [BEG1, GGOR, GoSt] among others. Our duaity theorem indicates a new route to those questions via a carefu study of LY and L and makes the study of these agebras more reevant and interesting. On one hand, the rationa Cherednik agebra is simper than the DAHA which, a priori, makes it ook ess appeaing. On the other hand, there are severa interesting features in the rationa case that do not have counterparts at east for the moment for the DAHA. We hope that the same can be said about Yangian-deformed doube oop agebras and quantum toroida agebras, whose representation theory is sti very mysterious. The former have a simper structure and one can hope that this wi make them easier to study and that it wi have some specia, interesting features that do not exist for toroida quantum agebras. 1

2 The trigonometric resp. rationa DAHA is generated by two subagebras, one isomorphic to a degenerate affine Hecke agebra and the other one isomorphic to the group agebra of an affine Wey group resp. both isomorphic to the smash product of S with a poynomia ring. For this reason, and because of the resuts mentioned above, we can expect its Schur-Wey dua to be buit from one copy of the Yangian Y for s n and from one copy of the oop agebra Ls n resp. two copies of s n C C[u]. This is indeed true for LY resp. L. 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 agebra considered in that paper is not described in a very precise way and no equivaence of categories is estabished. One motivation for the present artice comes from our desire to find exacty the reations between the generators of the Schur-Wey dua of a Cherednik agebra of type g. In the next three sections, we define Cherednik agebras and Yangians and expore some of their basic properties, in particuar their connections with doube affine Hecke agebras and quantum toroida agebras. The fifth section states the main resut theorem 5. for the trigonometric case, which is proved in the foowing one. After that, we ook more cosey at the action of certain eements of LY since this is usefu in the ast section, which concerns the rationa case theorem 8.1. Most of our resuts in the rationa case foows from the observation that the rationa Cherednik agebra of type g is contained in the trigonometric one. Our resuts are first proved for Cherednik agebras of type g, but we are abe to obtain simiar ones in type A 1 aso. Furthermore, our equivaence restricts to an equivaence between two categories of BGG-type theorem 8.3. Acknowedgments During the preparation of this paper, the author was supported by a Pionier grant of the Netherands Organization for Scientific Research NWO. He warmy thanks E. Opdam for his invitation to spend the year 005 at the University of Amsterdam. He aso thanks I. Cherednik, P. Etingof, V. Ginzburg, I. Gordon and T. Nevins for their comments. Hecke agebras and Cherednik agebras The definitions given in this section coud be stated for any Wey group W. However, in this paper, we wi be concerned ony with the symmetric group S, so we wi restrict our definitions to the case W = S. We set h = C. The symmetric group S acts on h by permuting the coordinates. Associated to h are two poynomia agebras: C[h] = Symh = C[x 1,..., x ] and C[h ] = Symh = C[y 1,..., y ], where {x 1,..., x } and {y 1,..., y } are dua bases of h and h, respectivey. For i, we set α i = x i x, αi = y i y, R = {α i 1 i } and R = {α i 1 i < }. The set Π = {x i x i1 1 i n 1} is a basis of simpe roots. The refection in h with respect to the hyperpane α = 0 is denoted s α, so s α y = y α, y α, where, : h h C is the canonica pairing. We set s i = s αi. The finite Hecke agebra H q associated to S is a deformation of the group agebra C[S ] and the affine Hecke agebra H q is a deformation of the group agebra of the extended affine Wey group S = P S where P is the attice i=1 Zx i h so C[ S ] = C[x ± 1,..., x± ] S. The agebra H q admits a degenerate form H c first introduced by Drinfed [Dr1] and by Lusztig [Lu]. Definition.1. The degenerate affine Hecke agebra H c of type g is the agebra generated by the

3 poynomia agebra Symh = C[z 1,..., z ] and the group agebra C[S ] with the reations s α z s α zs α = c α, z z h, α Π The doube affine Hecke agebra H q,κ defined in section introduced by I. Cherednik [Ch] aso admits degenerate versions: the trigonometric one and the rationa one. Reca that the group S is generated by s α α R and by the eement π = x 1 s 1 s 3 s 1,. Definition. Cherednik. Let t, c C. The degenerate trigonometric doube affine Hecke agebra of type g is the agebra H t,c generated by the group agebra of the extended affine Wey group C[ S ] and the poynomia agebra C[z 1,..., z ] = Symh subect to the foowing reations: s α z s α zs α = c α, z z h, α Π πz i = z i1 π, 1 i 1 πz = z 1 tπ Remark.1. The subagebra generated by C[S ] and the poynomia agebra C[z 1,..., z ] is isomorphic to the degenerate affine Hecke agebra H c. Definition.3. Let t, c C. The rationa Cherednik agebra H t,c of type g is the agebra generated by C[h], C[h ] and C[S ] subect to the foowing reations: w x w 1 = wx w y w 1 = wy x h, y h [y, x] = yx xy = t y, x c α R α, y x, α s α Remark.. The rationa Cherednik agebra H t,c of type A 1 is the subagebra of H t,c generated by C[x i x ] C[x 1,..., x ], by C[y i y ] C[y 1,..., y ] and by C[S ]. The exists a simpe reation between H t,c and H t,c. Proposition.1 [Su]. The agebra C[x ± 1,..., x± ] C[h] H t,c is isomorphic to H t,c. Before giving a proof of this proposition, we need to introduce eements in H t,c which wi be very usefu ater. For 1 i, set U i = t x iy i c <i s i and Y i = U i c i sign is i = t x iy i c i s i. Proposition.. [DuOp],[EtGi] 1. Y i = 1 x iy i y i x i.. U i U = U U i for any i,. 3. w Y i w 1 = Y wi.. The eements U i, 1 i, and C[S ] generate a subagebra of H t,c isomorphic to the degenerate affine Hecke agebra H c. 3

4 Remark.3. The eements Y i are not pairwise commutative if c 0: [Y, Y k ] = c [s i, s k ]. i=1 i,k Proof. The first statement foows from the equaity y i x i x i y i = t c x i x, y i x i, y i y s i = t c s i. =1, i =1, i The second part is proved in [DuOp]. The third part is obvious, so we prove ony the fourth one. If k i > 1, then s k,k1 U i = U i s k,k1, so the non-trivia reations that we have to check invove s i1,i and s i,i1 : s i1,i U i = t x i1y i1 s i1,i c <i1 s i,i1 U i = t x i1y i1 s i,i1 c <i1 s i1, s i1,i c = U i1 s i1,i c s i1, s i1,i c = U i1 s i,i1 c These two equaities, combined with the PBW-property of H t,c [EtGi] and of H c, compete the proof of part. In the proof of the two main theorems, we wi need the foowing identities. Proposition If i, then [y, x i ] = cs i and [x 1 i, y ] = cx 1 i x 1 s i.. [y i, x i ] = t c k i s i and [x 1 i, y i ] = tx i c i x1 i x 1 s i. 3. If i, then [Y, x i ] = c x i x s i and [x 1 i, Y ] = c x1 i x 1 s i.. [Y i, x i ] = tx i c i x i x s i and [x 1 i, Y i ] = tx 1 i c i x1 i x 1 s i. Proof. These are a immediate consequences of the definition of H t,c. Proof. of proposition.1. See aso [Su]. Because of proposition., part, and the PBW-property of H t,c and H t,c, we ony have to check the reation invoving π in definition.. First, assume that i. πu i = x 1 s 1 s 1, U i = x 1 U i1 cs 1,i1 s 1 s 1, = [x 1, U i1 ] U i1 x 1 cx 1 s 1,i1 s 1 s 1, = cx i1 s 1,i1 c[x 1, s 1,i1 ] U i1 x 1 cx 1 s 1,i1 s1 s 1, = U i1 x 1 s 1 s 1, = U i1 π

5 If i =, we obtain: πu = x 1 U 1 c s 1, s 1 s 1, = x 1 [x 1, y 1 ] cx 1 s 1, U 1 x 1 s1 s 1, = = = x 1 t c s i,1 cx 1 i 1 s 1, U 1 x 1 s1 s 1, = U 1 tπ = Coroary.1 of proposition.1. The agebra H t,c can aso be defined as the agebra generated by the eements x ± 1,..., x±, Y 1,..., Y and S with the reations w x i w 1 = x wi w Y i w 1 = Y wi [Y, Y k ] = c s k s ik s k s i i=1 i,k Y x i x i Y = tδ i x i c α, y x i, α x i s α s α x i. α R 3 Finite and oop Yangians The Yangians of finite type are quantum groups, introduced by V. Drinfed in [Dr1], which are quantizations of the enveoping agebra of the poynomia oop agebra g C C[u] of a semisimpe Lie agebra g. The second definition in [Dr] is given in terms of a finite Cartan matrix. If we repace it with a Cartan matrix of affine type, we obtain agebras that we ca oop Yangians LY β,λ. Let C n1 = c i 1 i, n1 Ĉn1 = c i 0 i, n1 be a Cartan matrix of finite resp. affine type A n1 resp. A 1 n1. If n 3: Ĉ n1 = Definition 3.1. [Dr], [ChPr] Let λ C. The Yangian Y λ associated to C n1 is the agebra generated by X ± i,r, H i,r, i = 1,... n 1, r Z 0, which satisfy the foowing reations : [H i,r, H,s ] = 0, [H i,0, X ±,s ] = ±c ix ±,s 1 [H i,r1, X ±,s ] [H i,r, X ±,s1 ] = ±λ c ih i,r X ±,s X±,s H i,r [X i,r, X,s ] = δ ih i,rs [X i,r ±, X±,s ] = 0 if 1 < i < n 1 3 5

6 [X ± i,r1, X±,s ] [X± i,r, X±,s1 ] = ±λ c ix ± i,r X±,s X±,s X± i,r [ X ± i,r1, [X ± i,r, X ±,s ]] [ X ± i,r, [X ± i,r 1, X ±,s ]] = 0 r 1, r, s 0 if i ±1 mod n 5 Remark 3.1. The Yangian Y λ1 is isomorphic to Y λ if λ 1 0 and λ 0. Definition 3.. Let β, λ C. The Yangian LY β,λ associated to Ĉn1 is the agebra generated by X ± i,r, H i,r, i = 0,... n1, r Z 0, which satisfy the reations of definition 3.1 for i, {0,..., n1} except that the reations, must be modified for i = 0 and = 1, n 1 in the foowing way when n 3: [H 1,r1, X ± 0,s ] [H 1,r, X ± 0,s1 ] = β λ λ H 1,rX ± 0,s λ λ βx± 0,s H 1,r 6 [H 0,r1, X ± 1,s ] [H 0,r, X ± 1,s1 ] = λ λ βh 0,rX ± 1,s β λ λ X± 1,s H 0,r 7 [H 0,r1, X ± n1,s ] [H 0,r, X ± n1,s1 ] = β λ λ H 0,rX ± n1,s λ λ βx± n1,s H 0,r 8 [H n1,r1, X ± 0,s ] [H n1,r, X ± 0,s1 ] = λ λ βh n1,rx ± 0,s β λ λ X± 0,s H n1,r 9 [X ± 1,r1, X± 0,s ] [X± 1,r, X± 0,s1 ] = β λ λ X± 1,r X± 0,s λ λ βx± 0,s X± 1,r 10 [X ± 0,r1, X± n1,s ] [X± 0,r, X± n1,s1 ] = β λ λ X± 0,r X± n1,s λ λ βx± n1,s X± 0,r 11 We wi aso impose the reation n1 i=0 H i,0 = 0. Remark 3.. We set X i ± = X i,0, H i = H i,0. If β = λ, the reations defining LY β,λ are the same as those in definition 3.1 with i, {0,..., n 1}. Note aso that the reations 6,7,8 and 9 a foow from 10 and 11 using reation 3; they were added above as a convenient reference since they wi be usefu ater in our computations. We shoud aso note that LY β1,λ 1 = LYβ,λ if β = ηβ 1 and λ = ηλ 1 for some η 0. When no confusion is possibe, we wi write LY and Y instead of LY β,λ and Y λ. Without the reation n1 i=0 H i = 0, we obtain the affine Yangian Ŷλ,β. Let = {ɛ i, 1 i n} be the root system of type A n1, E = span{ɛ i, i = 1,..., n}, ɛ i = ɛ i ɛ. We denote by, the non-degenerate biinear form on E given by ɛ i, ɛ = δ i. For a positive root ɛ, we denote by X ± ɛ the corresponding standard root vector of s n. If ɛ = ɛ i, i <, then X ɛ = E i and X ɛ = E i, where E rs is the matrix with 1 in the r, s-entry and zeros everywhere ese. In particuar, X θ = E 1n and X θ = E n1, where θ is the ongest root of s n. If ɛ = ɛ i,i1, then X ± ɛ = X ± i. One usefu 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 ony. The other eements are obtained inductivey by the formuas: X i,r1 ± = ±1 [H i,1, X i,r ± ] 1 H ix i,r ± X± i,r H i, H i,r1 = [X i,r, X i,1 ]. 1 Furthermore, the subagebra generated by the eements with r = 0 is a quotient of actuay, is isomorphic to the enveoping agebra of the Lie resp. oop agebra s n resp. Ls n = s n C 6

7 C[u, u 1 ]. The subagebra Yλ 0 Y λ. generated by the eements with i 0 is a quotient of the Yangian The two subagebras Yλ 0 and ULs n generate LY β,λ. Indeed, combining the observations in the previous two paragraphs, we see that we ony have to show that the subagebra they generate contains X 0,1 ±. From the reation 1 in definition 3. 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 λ λ H 1X ± 0 λ λ βx± 0 H 1. Thus X ± 0,1 hence aso H 0,1 beongs to the subagebra of LY β,λ generated by Y 0 λ and ULs n. For 1 i n 1, set JX ± i = X± i,1 λω± i and ω ± i = ω ± i,1 ω± i, where ω ± i,1 = ±1 ɛ [X ± i, X ± ɛ ]X ɛ X ɛ [X ± i, X± ɛ ], ω ± i, = 1 X± i H i H i X ± i JH i = H i,1 λν i where ν i = 1 ɛ, ɛ i,i1 X ɛ Xɛ Xɛ X ɛ 1 H i. ɛ More expicity, since X i = E i,i1, Xi = E i1,i and H i = E ii E i1,i1 for 1 i n 1, we can write ω i = 1 n sign ie,i1 E i E i E,i1 1 E i,i1h i H i E i,i1 13 ω i = 1 =1 i,i1 n =1 i,i1 sign ie i1, E i E i E i1, 1 E i1,ih i H i E i1,i 1 It is possibe to define eements Jz Y for any z s n in such a way that [Jz 1, z ] = J[z 1, z ]: this foows from the isomorphism given in [Dr] between two different reaizations of the Yangian Y λ : the one given above and the one first given in [Dr1] in terms of generators z, Jz z s n the Jz s satisfy a deformed Jacobi identity. In the proof of our first main theorem, the foowing agebra automorphism wi be very important. Lemma 3.1. It is possibe to define an agebra automorphism ρ of LY by setting r r λ rs r ρh i,r = H s i1,s, ρx i,r ± r λ rs = X s i1,s ± for i 0, 1 s=0 ρh i,r = r s=0 r s s=0 r β rs H i1,s, ρx i,r ± r = s s=0 β rs X ± i1,s for i = 0, 1 We use the convention that X 1,r ± = X± n1,r and H 1,r = H n1,r. Note that, in particuar, ρx i ± = X i1 ±, ρh i = H i1 i and ρx i,1 ± = X± i1,1 λ X± i1, ρh i,1 = H i1,1 λ H i1 if i 0, 1, whereas ρx i,1 ± = X± i1,1 βx± i1, ρh i,1 = H i1,1 βh i1 if i = 0, 1. The automorphism ρ is very simiar to the automorphism τ λ or τ β in [ChPr] foowed by a decrement of the first index. 7

8 Proof of emma 3.1. We have to verify that ρ is indeed an automorphism of LY, that is, that it respects the defining reations of LY. In the case when i, 0, 1 in the reations 1-5, this foows from the fact that ρ is the same as the automorphism τ λ from [ChPr] foowed by a decrement of the indices. A short verification shows that ρ preserves the reations 1,3 and 5 when i = 0, 1 or = 0, 1. In the case of equation 3 and i =,, one has to use the identity r s r s ab=k =. Since the reations 6-9 foow from 10 and 11 by a b k appying [, X?,0],? = 0, 1, n 1, there are three cases eft that require a more detaied verification. r r 1 r 1 We wi use the identity =. a a a 1 Case 1: With i =, = 1 in reation, we find that ρ [X ±,r1, X± 1,s ] [X±,r, X± 1,s1 ] is equa to = = = = = r1 s a=0 b=0 s1 r1 r a=0 b=0 s a=0 b=0 s1 a=0 b=0 r a=0 b=0 r 1 a r a r a s b s 1 b r a 1 r r s a b s r s a b r1 s a=0 b=0 s1 a=0 b=0 r a 1 λ s b [ ] λ r1a X 1,a ±, βsb X ± 0,b [ λ ra X 1,a ±, βs1b X ± 0,b s b s b 1 λ r1a β sb [X 1,a ±, X± 0,b ] λ ra β s1b [X 1,a ±, X± 0,b ] ra β sb λ β[x± 1,a, X± 0,b ] λ ra1 β sb [X 1,a ±, X± 0,b ] λ ra β sb1 [X 1,a ±, X± 0,b ] r r s a b 1 a=0 b=0 r s r s λ ra β sb λ a b β[x± 1,a, X± 0,b ] a=0 b=0 r s r s λ rã β sb [X ã b 1,ã1 ±, X± 0,b ] ã=0 b=0 r s r λ ra β a s b s b[x 1,a ±, X± ] 0, b1 r s r s λ ra β sb λ a b β[x± 1,a, X± 0,b ] a=0 b=0 r s r λ ã s b ã=0 b=0 rã β s b [X ± 1,ã1, X± 0, b ] [X± 1,ã, X± 0, b1 ] 8 ]

9 = = r s ã=0 b=0 = ρ r ã=0 b=0 r ã=0 b=0 r ã s r ã λ rã β s b λ β[x± 1,ã, X± ] 0, b λ rã β s b s b β λ λ X± 1,ã X± λ 0, b λ βx± 0, b X± 1,ã λ rã β s b λ X 1,ã ± X± 0, b X± 0, b X± 1,ã s b s r ã s b X ±,r X±1,s X±1,s X±,r λ Case : i = 1, = 0. We have to prove that ρ [X ± 1,r1, X± 0,s ] [X± 1,r, X± 0,s1 ] = ρ β λ λ X± 1,r X± 0,s λ λ βx± 0,s X± 1,r. This case is anaogous to case 1, but a itte bit simper since the term λ β β = 0. β above becomes Case 3: i = 0, = n 1. We have to show that ρ [X ± 0,r1, X± n1,s ] [X± 0,r, X± n1,s1 ] = ρ β λ λ X± 0,r X± n1,s λ λ βx± n1,s X± 0,r. The computations are again very simiar to those of case 1: the main difference is that the factor λ β gets repaced by β λ. Reations with DAHA s and toroida quantum agebras It is known that the Yangians of finite type can be obtained from quantum oop agebras via a imiting procedure [Dr3] and that the same is true about the trigonometric Cherednik agebra and the doube affine Hecke agebra or eiptic Cherednik agebra, see [Ch3] for instance. We wi reca these resuts and expain how the oop Yangians introduced in section 3 can be obtained from toroida quantum agebras. Definition.1 Cherednik. Let q, κ C. The doube affine Hecke agebra H q,κ of type g is the unita associative agebra over C with generators T i ±1, X ±1, Y ± for i {1,..., 1} and {1,..., } satisfying the foowing reations: where X 0 = X 1 X X. T i 1T i q = 0, T i T i1 T i = T i1 T i T i1 T i T = T T i if i > 1, X 0 Y 1 = κy 1 X 0, X Y 1 1 X 1 Y 1 = q T 1 X i X = X X i, Y i Y = Y Y i, T i X i T i = q X i1, Ti 1 Y i Ti 1 = q Y i1, X T i = T i X, Y T i = T i Y if i, i 1 9

10 Remark.1. We set y = 1 in definition 1.1 in [VaVa1]. The trigonometric Cherednik agebra can be viewed as a imit degenerate version of the doube affine Hecke agebra. We sketch here a few computations which iustrate this fact. We extend the scaars from C to C[[h]] and consider the competed agebra H q,κ [[h]] with q = e c h, κ = e th. Setting Y i = e hu i, the equaity X Y1 1 X 1 Y 1 = q T1 becomes X 1 hu 1 X 1 1 hu 1 = 1 chcht 1 1 ch oh, where oh is in h H q,κ [[h]]. Canceing the constant term 1 on both sides, dividing by h and then etting h 0 gives x U 1 x 1 U 1 = cs 1, which impies that [U 1, X ] = cs 1 x. In this imit, the finite Hecke agebra identifies with C[S ] and T 1 with s 1. This is indeed the reation between U 1 and x in H t,c, as can be seen from the third reation in proposition.3 with = 1, i =. If we make the same substitution in the reation X 0 Y 1 = κy 1 X 0, we obtain X 1 X 1hU 1 = 1 th1 hu 1 X 1 X oh. Subtracting X 1 X on both sides, dividing by h and etting h 0 gives [U 1, x 1 x ] = tx 1 x, which impies that k=1 x 1 x k1 [U 1, x k ]x k1 x = tx 1 x. Since [U 1, x k ] = cs 1k x k for k 1, [U 1, x 1 ]x x c k= s 1kx x = tx 1 x, which eads to [U 1, x 1 ] = tx 1 c k= s 1kx k : after some simpifications, we obtain the fourth reation in proposition.3. The rationa, trigonometric and eiptic Cherednik agebras of S are a isomorphic after competion: see [Ch3] for a detaied discussion. This impies that, for generic vaues of the deformation parameters, these three agebras have equivaent categories of finite dimensiona representations. For modues which are not finite dimensiona, we don t have such an equivaence in genera. However, it is sometimes possibe to ift a modue over H or H to one over H if the parameters satisfy certain technica conditions: see [Ch3] section.1 for more on this subect. Definition.. Let q 1, q C. The toroida quantum agebra Üq 1,q of type A n1 is the unita associative agebra over C with generators e i,r, f i,r, k i,r ki,0 1, i {0,..., n 1}, r Z which satisfy the foowing reations: [k i,r, k,s ] i, {0,..., n 1}, r, s Z 15 k i,0 e,r = q c i 1 e,rk i,0, k i,0 f,r = q c i 1 f,r k i,0, q 1 q 1 1 [e i,r, f,s ] = δ i k i,rs k i,rs 16 Here, k ± i,rs = k i,rs if ±r s 0 and = 0 otherwise. The next three reations hod i, {0,..., n 1}, r, s Z except for {i, } = {n 1, 0}, {0, 1}: k i,r1 e,s q c i 1 k i,re,s1 = q c i 1 e,sk i,r1 e,s1 k i,r 17 e i,r1 e,s q c i 1 e i,re,s1 = q c i 1 e,se i,r1 e,s1 e i,r 18 {e i,r e i,s e,t q 1 q 1 1 e i,re,t e i,s e,t e i,r e i,s } {r s} = 0 if i ±1 mod n 1 19 The same reations hod with e i,r repaced by f i,r and q c i 1 by q c i 1. In the cases {i, } = {n1, 0}, {0, 1}, we must modify the reations above in the foowing way: we introduce a second parameter q in such a way that we obtain an agebra isomorphism Ψ of Üq 1,q given by e i,r, f i,r, k i,r q1 re i1,r, q1 rf i1,r, q1 rk i1,r for i n 1 and e i,r, f i,r, k i,r 10

11 q r e i1,r, q r f i1,r, q r k i1,r if i = 0, 1. We identify e 1,r with e n1,r, etc. For instance, reation 18 for i = 0, = 1 becomes q e 0,r1 e 1,s e 0,r e 1,s1 = q 1 1 q e 1,s e 0,r1 q 1 e 1,s1 e 0,r, and with i = n 1, = 0 we have the very simiar identity: q e n1,r1 e 0,s e n1,r e 0,s1 = q 1 1 q e 0,s e n1,r1 q 1 e 0,s1 e n1,r. Remark.. We coud have expressed the reations above and aso those for Yangians using power series as in [VaVa1]. The definition in [VaVa1] invoves a centra parameter c which we have taken to be equa to 1. The subagebra U hor q 1,q generated by the eements e i,0, f i,0, k i,0 ±1, i {0,... n1} is a quotient the quantum affine agebra of type Ân1. The subagebra U ver q 1,q generated by the eements e i,r, f i,r, k i,r, ki,0, i {1,... n 1}, r Z is a quotient of the quantum oop agebra of type  n1. The connection between the representation theory of the quantum affine resp. toroida agebras and the Yangians of finite resp. affine type is ess direct than in the case of Hecke agebras. However, in view of the reation between Ü and LY expained beow, which is an extension of a resut of Drinfed in the finite case, one can often expects that resuts which are true for quantum affine or toroida agebras have anaogs for Yangians which can be proved using simiar arguments. It is known that the Yangians of finite type and the quantum affine agebras have the same finite dimensiona representation theory: this was proved using geometrica methods in [Va]. More genera equivaences are not known at the moment between these two types of agebras. It is possibe to view the Yangian Y λ as a imit version of the quantum affine agebra U q. The same is true for Ŷλ,β and Üq 1,q. Let Ü[[h]] be the competed agebra with parameters q 1 = e λ h, q = e βh. h 0 Consider the kerne K of the composite map Ü[[h]] Üh=0 Uŝ n = U hor h=0. Let A be the C[[h]]-subagebra of Üh generated by Ü[[h]] and K h. Then the quotient A/hA is isomorphic to Ŷλ,β. To see this, et A 1 be the subagebra of A generated by U ver K U ver and h. Since U ver is a quotient of the quantum oop agebra, A 1 /ha 1 is a quotient of the Yangian Y λ see [Dr3], that is, we have an epimorphism ζ : Y λ A 1 /ha 1. The automorphism Ψ of Ü[[h]] induces an automorphism, aso denoted Ψ, on A. It is reated to the automorphism ρ of Ŷλ,β in the foowing way for i n: ΨζX i,r ± = ζρx± i,r, ΨζH i,r = ζρh i,r Ψ ζx ± 1,r = ζρ X ± 1,r, Ψ ζh 1,r = ζρ H 1,r From these reations, one concudes that it is possibe to extend ζ to Ŷλ,β by setting ζx ± 0,r = Ψζρ 1 X ± 0,r and simiary for H 0,r. One can show that we have thus obtained an isomorphism. 5 Schur-Wey duaity functor The Schur-Wey duaity estabished by M. Varagnoo and E. Vasserot [VaVa1] invoves, on one side, a toroida quantum agebra and, on the other side, a doube affine Hecke agebra for S. Theorem 11

12 5. estabishes a simiar type of duaity between the trigonometric DAHA H t,c and the oop Yangian LY β,λ, which extends the duaity for the Yangian of finite type due to V. Drinfed [Dr1]. Before stating the more cassica resuts on the theme of Schur-Wey duaity, we have to define the notion of modue of eve over s n and over the quantized enveoping agebra U q s n. Fix a positive integer n and set V = C n. Definition 5.1. A finite dimensiona representation of s n or U q s n q not a root of unity is of eve if each of its irreducibe components is isomorphic to a direct summand of V. Theorem 5.1. [Ji, Dr1, ChPr1] Fix 1, n and assume that q C is not a root of unity. Let A be one of the agebras C[S ], H q S, H c=1 S, H q S, and et B be the corresponding one in the same order among Us n, U q s n, Y λ=1 s n, U q Ls n. There exists a functor F from the category of finite dimensiona right A-modues to the category of finite dimensiona eft B-modues which are of eve as s n -modues in the first and third case and as U q s n -modues in the second and fourth case which is given by FM = M C V where C = C[S ] first and third case or C = H q S second and fourth case. Furthermore, this functor is an equivaence of categories if n 1. The s n modue structure on V commutes with the S -modue structure obtained by simpy permuting the factors in the tensor product. Let M be a right modue over H t,c. Since C[S ] H t,c, we can form the tensor product FM = M C[S ] V. On one hand, since H t,c contains the degenerate affine Hecke agebra H c, M can be viewed as a right modue over H c, so it foows from [Dr1] that FM is a modue of eve over the Yangian Y λ of s n with λ = c. On the other hand, H t,c aso contains a copy of the group agebra of the extended affine Wey group S, so it foows from [ChPr1] the case q = 1 that FM is aso a modue of eve over the oop agebra Ls n. These two modue structures can be gued together to obtain a modue over LY. This is the content of our first main theorem. Before stating it, we need one definition. Definition 5.. A modue M over LY λ,β or over L λ,β is caed integrabe if it is the direct sum of its integra weight spaces under the action of h and if each generator X i,r ± acts ocay nipotenty on M. Theorem 5.. Suppose that 1, n 3 and set λ = c, β = t nc c. The functor F : M M C[S ] V sends a right H t,c -modue to an integrabe LY β,λ -modue of eve as s n -modue. Furthermore, if < n, this functor is an equivaence. Remark 5.1. This theorem is very simiar to the main resut of [VaVa1] where H t,c is repaced by a doube affine Hecke agebra and LY β,λ is repaced by the toroida quantum agebra Üq 1,q it is defined sighty differenty in op.cit, under the assumption that the parameter q 1 is not a root of unity. However, theorem 5. is not an immediate consequence of the DAHA case in op.cit. since, in genera, we don t have equivaences between categories of modues over H and H or over LY and Ü. The first part of the proof of theorem 5. can be deduced from a proposition in [VaVa1]: this is expained in detai in the next section. However, the fact that the functor F is essentiay surective must be given an independent but simiar proof for the aforementioned reason. The same is true for the rationa case treated in section 8. 1

13 6 Proof of theorem 5. The proof of theorem 5. consists of two parts. First, we show how to obtain an integrabe LY β,λ - modue structure on FM, and then we prove that any integrabe representation of LY β,λ of eve is of the form FM. If there is no confusion possibe for the vaues of the parameters, we wi write H, H, H, H, LY instead of H q,κ, H t,c, H c, H t,c, LY β,λ. 6.1 Proof of theorem 5., part 1 Fix m M, v = v i1 v i V, where {v 1,..., v n } is the standard basis of C n and 1 i n. The subagebra s n generated by the eements X ± i, H i, 1 i n 1, acts on V as usua. The eement z u ± Ls n acts on FM in the foowing way: z u ±k m v = =1 mx ±k v i1 zv i v i. For z s n, we wi write z v for v i1 zv i v i. The eements JX ± i, JH i and X ± i,1, H i,1, 1 i n 1, act on FM in the foowing way see [Dr1],[ChPr]: JX ± i m v = =1 my X ±, i v, X ± i,1 m v = JX± i m v λω± i m v, 0 JH i m v = my H i v, H i,1m v = JH i m v λν i m v. 1 =1 The foowing observation wi be very usefu: the action of s k on V if given in terms of matrices by: s k = n r, E rsesr. k It is possibe to give another, somewhat simper formua for the action of X ± k,1 and H k,1 if we assume that i 1 i... i. We wi denote by k resp. k the first resp. ast vaue of such that i = k and we set k = k k 1. We wi adopt the foowing notation: v = v i1 v v ik1 1 k v v ik1 1 i or v k1 = 0 if i k 1 for any 1 ; k1 v k = v i1 v v i k 1 k1 v v i k 1 i or v k 1 k n 1, we have: = 0 if i k for any 1. For k1 X k,1 m v = m d= k1 s d,k1 Uk1 λ n k v k1 X k,1 m v = m k d= k s d, k U k λ n k v k 13

14 H k,1 m v = k d= k s d,k Uk m v k1 d= k1 s d,k1 Uk1 m v n k λ k k1 m v λ k k k1 1m v We prove ony the identity for X k,1, the other cases being simiar. Suppose that i 1 i i. We compute: ω n k k,1 v = = = n k k1 k1 d= k1 E d k,k1 v 1 k1 d= k1 s dk1 s dk1 v k1 1 d= k1 n k 1 k1 By s dd, we mean simpy the identity eement in S. d= k1 a=1 =1 a k,..., k1 k1 n sign kek a Ed,k1 v signa k1s ade d= k1 a=1 a k,..., k1 d k,k1 v signa k1s ak1 v 3 k1 a=1 a k,..., k1 Notice that and that k1 k1 k1 ω k, v = 1 k k1 1Ek,k1 d v d= k1 d= k1 s dk1 k1 k s dk1 d= k1 a= k1 1 k1 s ak1 v = k Ek,k1 d v k1 a= k d= k1 s ak1 v k1 = k1 1 k1 d= k1 E d k,k1 v Putting equaities 0, 3 and together gives us: k1 X k,1 m v = m s d,k1 Y k1 λ n k signa k1s ak1 v k1 d= k1 a=1 a k1 This is the formua for the action of X k,1 on m v. Remark 6.1. For 1, we define eements in S by s 1, = s 1, 1 1s 1 1, 1 s 1, and s, 1 = s, 1s 1, s 1 1, 1. Then, in formua, we can repace s d,k1 by s d,k1 in the -case and by s d, k in the case: one has to notice that we can make this substitution in our computations above. 1

15 Foowing one of the main ideas in [VaVa1], we define a inear automorphism T of M C[S ] V in the foowing way: T m v i1 v i = mx δ i 1,n 1 x δ i,n v i1 1 v i 1, with the convention that v n1 = v 1. Here, δ i, is the usua deta function. We set v 1 = v i1 1 v i 1. One can check that T ϕx ± i1 = ϕx± i T and T ϕh i1 = ϕh i T for any 0 i n 1, where ϕ : Y End C FM is the agebra map coming from the Y -modue structure on FM. Reca the automorphism ρ from section 3. The foowing emma wi be crucia. Lemma 6.1. Let M be a modue over H. Suppose that λ = c and β = t nc c. For any i n 1 and any r 0, the foowing identities between operators on FM hod: ϕ ρx ± i,r = T 1 ϕx ± i,r T ϕ ρh i,r = T 1 ϕh i,r T 5 ϕ ρ X ± 1,r = T ϕx ± 1,r T ϕ ρ H 1,r = T ϕh 1,r T 6 There are two ways to prove this emma. One is to deduce it from proposition 3. in [VaVa1] using the fact that the trigonometric Cherednik agebra is a imit version of the doube affine Hecke agebra. The second one is by direct computations. We wi start with the first approach and afterwards we wi give a sketch of the reevant computations. Proof. We can restrict ourseves to proving emma 6.1 when M = H. Since the eements X i,r ±, H i,r with r = 0, 1 generate LY see equation 1, it is enough to prove the emma for r = 0, 1. First, we prove reation 5 for X i,1 with i n 1. The proof for X i,1 is exacty the same and we omit it, and the proof for H i,1 foows from either of these two cases using identity 3. We start with proposition 3. in [VaVa1] in the case M = H q,κ. We choose a v as before and assume that i 1 i i. The aforementioned proposition, aong with theorem 3.3 in oc. cit., says that we have the foowing identities in End C H H V for k {,..., n 1} concerning the action of q n e k,1 e k,0 and q n e k1,1 e k1,0 on 1 v for k n 1: q 1 k X 1 n q 1 k 1 X 1 k 1 d= k T d,k 1 Here, for d k, T d,k = T d T d1 T k. k 1 d= k T d,k q n q nk Y k 1 1 v k 1 = 7 q n q nk1 q 1 Y k 1 1 X 1 n X 1 v k 1 Now we extend the base ring by repacing C by C[[h]] and set q = e ch, κ = e th. We view both sides of identity 7 as eements of H[[h]] H[[h]] V [[h]]. Let us denote by a : H[[h]] H[[h]] an isomorphism between these two competed agebras as described in [Ch3]. Such an isomorphism can be obtained from a study of intertwiners. In particuar, we have ay i = e hu i and, in the 15

16 quotient H = H[[h]]/hH[[h]], at i s i,i1, ax i x i. Using a, we can identify H[[h]] H[[h]] V [[h]] with H[[h]] C[S ][[h]] V [[h]]. After canceing q 1 k in 7, we appy the isomorphism a and equate the coefficients of h on both sides. We then obtain x 1 n x 1 1 k d= k 1 1 s d,k k d= k 1 s d,k U k c U k c n k v k 1 = n kx1 n x 1 v k 1 8 The equation 5, which we want to prove, says that for the case of X k,1, k n 1, and M = H, m = 1: k d= k s d,k U k λ n k x 1 n x 1 v k 1 λ x 1 n x 1 This is equation 8 since c = λ. k d= k s d,k k d= k s d,k U k λ n k v k 1 As for equation 6 in the case of X 1,1 and M = H, m = 1, it says that β d= n s d,n d= n s d,n U n λ n n x 1 n1 x 1 v n = x 1 n1 x 1 v n x1 n1 x 1 Since β = t nλ λ, this is equivaent to d= n s d,n d= n s d,n U n tx 1 n1 x 1 v n = x1 n1 x 1 x 1 n x 1 v k 1 = U n λ n v n. d= n s d,n U n v n. 9 This can aso be deduced from proposition 3. in [VaVa1] with M = H. Indeed, this proposition says that we have the foowing reation concerning the action of e 1,1 e 1,0 and of e n1,1 e n1,0 on 1 v: q 1n X 1 n1 X d= n T d,n q n1 κy n 1 1 v n = 16

17 q 1n 1 1 d= n T d,n q nn1 q n Y n 1 1X 1 n1 X 1 v n. We proceed as previousy: we cance q 1n on both sides, extend the base ring to C[[h]], set q = e ch, κ = e th. We then appy the isomorphism a and equate the coefficients of h; this yied x 1 n1 x 1 cn 1 U n t v n = d= n s d,n d= n s d,n U n After a simpe simpification, we obtain equation 9. cn 1 x 1 n1 x 1 v n. Proof of emma sketch of aternative approach. To simpify the notation, wi not use ϕ in the proof. We used it ony to state the emma in a convenient way. We have to check the equaity JX i λω i T m v = T JX i1 λω i1 λ X i1 m v. 30 With v as before but without assuming that i 1 i i, suppose that 1 < < p are exacty the vaues of for which i = n. Then T m v = mx 1 1 x 1 p x 1 1 x 1 p. v 1. Set x 1 1,..., p = JX i T m v = = k=1 p mx 1 1,..., p Y k E k i,i1v 1 r=1 k=1 k s s k=1 mx 1 1 x 1 r1 [x 1 r, Y k ]x 1 r1 x 1 p Ei,i1v k 1 31 my k x 1 1,..., p E k i,i1v 1 3 and k=1 my k x 1 1,..., p E k i,i1v 1 = T JX i1 m v 33 Therefore, we must prove that 31 λω i T m v = λt ω i1 m v λ T X i1 m v 3 17

18 To compute the action of ω i on v 1, we distinguish two cases: when E,i1 and E i act on the same tensorand, and when they act on different ones. ω i n i T m v = mx 1 1,..., p Ei,i1v k n i 1 = T m E i1,iv k 35 k=1 k=1 1 n =1 k=1 d=1 i,i1 i k =1 i d =i sign imx 1 1,..., p E k i,i1 skd v mx1 1,..., p E i,i1 H i H i E i,i1 v 1 37 Doing the same for ω i1, we obtain that ω i T m vt ω i1 m v 1 T X i1 m v equas 1 =1 k=1 d=1 i,i1 i k =1 i d =i T 1 n =1 k=1 d=1 i1,i i k = i d =i sign imx 1 1,..., p E k i,i1s kd v1 38 sign i 1m Ei1,i k skd v 39 Therefore, the equaity 3 that we have to prove simpifies to 31 = By considering the two different cases: 1 and = 1 in 38 and n, = n in 39, we find the foowing expression for 38 39, which equas 31 using [x 1 r, Y k ] = c x1 r x 1 k s k r : skd v 1 1 T skd v = 1 k=1 d=1 i k =n i d =i mx 1 1,..., p E k i,i1 k=1 d=1 i k =n i d =i m E k i1,i We now prove the case i = 1 in emma 6.1. Suppose that 1,..., p resp. γ 1,..., γ e are exacty the vaues of resp. of γ such that i = n resp. i γ = n 1. JX 1,0 T m v = p e mx 1 1,..., p x 1 γ 1 u=1 p p r=1 k=1 x 1 γ u1 [x 1 γ u, Y s ]x 1 γ u1 x 1 γ e E s 1 v 0 mx 1 1 x 1 r1 [x 1 r, Y s ]x 1 r1 x 1 p x 1 γ 1,...,γ e E s 1 v 1 my k x 1 1,..., p x 1 γ 1,...,γ e E k 1v The ast term is equa to T JX n1 m v. We can decompose 0 and 1 into severa sums using the reations [x 1 γ u, Y s ] = c x1 γ u x 1 s s γu,s and [x 1 r, Y r ] = tx 1 r c k=1 x 1 k s k r. 18 x 1 r k r

19 After some ong computations, we obtain: 0 1 = t p a=1 c c mx 1 1,..., p x 1 γ 1,...,γ e E a 1 v p mx a=1 q=1 q d,γ h p 1 1,..., a1 x 1 a x 1 a1,..., p x 1 γ 1,...,γ e E1 q sa,qv 3 1 mx 1,..., a1 x 1 q x 1 a1,..., p x 1 a=1 q=1 q d,γ h γ 1,...,γ e E q 1 sa,qv We now focus on ω 1 T m v and T ω n1 m v. By considering the cases when E 1, E and E n1,, E n act on the same tensorand and on different ones, we can write: ω 1 T m v T ω n1 m v = 1 n p =3 q=1 b=1 i q= 1 n T n =1 q=1 b=1 i q= mx 1 γ 1,...,γ e E q 1 sq,b v 5 p m E q n1,n sq,b v 6 T E n1,n m v 7 One can check that 3 = 5 and 6 =. Finay, we get X 1,1 T m v T X n1,1 m v = 0 1 λ5 λ6 λ7 = βt E n1,n m v. Using the emma, we can now define the action of X ± 0,1 and of H 0,1 on FM 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 emma 6.1 impies that X ± 0,1 T m v = T X ± n1,1 m v βx ± n1 m v and simiary for H 0,1. In other words, and more generay, we set ϕx ± 0,r = T ϕ ρx ± 0,r T 1, ϕh 0,r = T ϕ ρh 0,r T 1 r 0. We now have to check that this indeed gives FM a structure of integrabe modue over LY. Choose i,, k {0, 1,..., n 1} with k i, k. We have to verify that ϕx ± i,r, ϕh i,r, ϕx ±,s 19

20 and ϕh,s satisfy the defining reations of LY. This is true when k = 0 from theorem 1 of [Dr1]. Using emma 6.1, we concude that it is aso true for k 0. This means that we have a we-defined agebra homomorphism ϕ from LY to End C FM. That FM is integrabe foows from the fact that V is an integrabe s n -modue, and that it is of eve foows from theorem 5.1 in the case of C[S ] and Us n. 6. Proof of theorem 5., part For the rest of this section, we assume that < n so, in particuar, n. In the second step of the proof, we have to show that, given an integrabe modue M of eve over LY, we can find a modue M over H such that FM = M. Such an M cannot, in genera, be ifted to a modue over Üq 1,q, so this second step is not an immediate consequence of [VaVa1], athough the approach is simiar. Integrabe Us n -modues are direct sums of finite dimensiona ones, so, by the resuts of Drinfed [Dr1] and Chari-Pressey [ChPr1], we know that there exists modues M 1 and M over, respectivey, H and C[ S ], such that M = FM 1 as Y -modue and M = FM as Ls n -modue. Since C[S ] H and C[S ] C[ S ], we have an isomorphism M 1 = M of S -modues, so we can denote them simpy by M. We have to show that M is an H-modue. The foowing wi be usefu. Lemma 6.. If v = v i1 v i is a generator of V as a modue over Us n that is, if i i k for any k, then m v = 0 = m = 0. Fix 1, k, k. We choose v to be the foowing generator of V as Us n -modue: v = v i1 v i v i where i d = d 3 if d <, d k, i d = d if d >, d k, i = and i k = 1. We can express ω as an operator on V in the foowing way: n n ω V = 1 sign de3d r Es d E3 r 1 E r 3H s d=1 r=1 d,3 s r Therefore, [E a n1, ω ] = 1 r=1 gives r a r=1 r=1 s r E31 r Ea n 1 E31 a Es n and appying this to m v with a =, k s a [E n1, ω ]m v = 1 Ek 31E n m v and [Ek n1, ω ]m v = 1 Ek 31E n m v. X,1 X 0 X 0 X,1 m v = = r=1 mxr Y s X,s,0 Er n1v my s x r E r n1x,s,0 v λ[ω, X 0 ]m v m[x r, Y s ] E3E s n1v r λ r=1 s mx a [En1, a ω ]v a=1 = m[x k, Y ] E 3 Ek n1v λ mx E k 31E n v λ mx k E k 31E n v = m [x k, Y ] λ x x k s k ṽ 0

21 where ṽ = E 3 Ek n1 v. We know from reation 3 that [X,1, X 0 ] = 0, so the ast expression is equa to 0. Since ṽ is a generator of V as a Us n -modue, it foows, from emma 6. and our assumption that λ = c, that m [x k, Y ] c x x k s k = 0. We consider now the reation between x k and Y k. From the definition of ν 1 : ν 1 = 1 n E 1d E d1 E d1 E 1d 1 E 1E 1 E 1 E 1 1 d=3 whence, as an operator on V, it is equa to ν 1 V = 1 Therefore, [E r n1, ν 1 ] = 1 n d=3 =1 s n1 d=3 E 1d Es d1 E d Es d End r Es d1 1 s r =1 E 1 Es 1 1 s H0E r n1 s E1E r n s s r n E d E d E d E d 1 H 1 d=3 =1 H n 1 Hs 1 H 1. s EnE r 1 s s r =1 n En1H r 1 s s r E r n1. Fix k, 1 k. We now choose v to be equa to v = v i1 v i with i d = d if d < k, i d = d 1 if d > k and i k = 1. Note that i d, n, n 1 d since 1 < n 1 by assumption. Appying the previous expression for [En1 r, ν 1] to v, we obtain the foowing: [En1, r ν 1 ]v = 1 n End r Ek d1 v = 1 n s kren1v k if r k [En1, k ν 1 ]v = E k n1v. 8 d=3 We need 8 to obtain equation 9 beow. Note that H1,0 s v = 0 if s k. H 1,1 X 0 X 0 H 1,1m v = r=1 mx r Y s H1,0E s n1v r r=1 my s x r En1H r 1,0v s λ[ν 1, X 0 ]m v = my k x k E k n1h k 1,0v λ = my k x k E k n1v λ n λ = my k x k ṽ λ 1 mx r [En1, r ν 1 ]v r=1 mx r s kr En1v k r=1 r k m E k n1v n mx r s kr ṽ λ r=1 r k m ṽ 9

22 where ṽ = E k n1 v. We want to obtain a simiar reation with H 1,1 repaced by H n1,1. From the definition of ν n1, ν n1 = 1 n E dn E nd E nd E dn 1 E n1,ne n,n1 E n,n1 E n1,n d=1 n 1 E d,n1 E n1,d E n1,d E d,n1 1 H n1 d=1 whence, as an operator on V, it is equa to ν n1 V = 1 n d=1 =1 s 1 =1 E dn Es nd E d,n1 Es n1,d =1 H n n1 Hs n1 H n1. s =1 s E n1,n Es n,n1 Therefore, [E r n1, ν n1 ] = 1 n d= Ed1 r Es nd 1 s r En1,1E r n,n1 s s r H0E r n1 s En,n1E r n1,1 s s r n En1H r n1 s s r E r n1. Appying the previous expression for [En1 r, ν n1] to v, we concude that [En1 r, ν n1]v = 0 if r k and [En1, k ν n1 ]v = 1 n n Ed1 k Es nd v E k n1v = 1 n s ks E k n1v E k n1v d= s k This equation aows us to compute [H n1,1, X 0 ]m v: H n1,1 X 0 X 0 H n1,1m v = mxr Y s Hn1,0E s n1v r my s x r En1H r n1,0v s r, λ[ν n1, X 0 ]m v s k = mx k Y k H k n1,0e k n1v λ = mx k Y k ṽ λ s k mx r [En1, r ν n1 ]v r=1 mx k s ks n ṽ 50

23 From the reations 1, 6 and 9 in LY, we know that X 0,1 = [H 1,1, X 0 ] λ βh 1X 0 βx 0 H 1 51 = [H n1,1, X 0 ] βh n1x 0 λ βx 0 H n1 5 Appying these two expressions for X 0,1 to m v, using equaities 9,50 and the fact that H 1 X 0 v = 0 and X 0 H n1v = 0 because of our choice of v, we obtain: my k x k ṽ λ mx k Y k ṽ λ = m[x k, Y k ] ṽ λ n mx r s kr ṽ λ r=1 r k n mx k s ks ṽ λ s k n mx r x k s kr ṽ λ r=1 r k mx k ṽ βx 0 H 1m v = mx k ṽ βh n1 X 0 m v mx k ṽ βmx k ṽ = 0 Since ṽ is a generator of V as a Us n -modue, it foows from emma 6. and our assumptions that β λn = t, λ = c that m [x k, Y k ] c x r x k s kr tx k = 0 r=1 r k We proved above that m[x k, Y ] c x x k s k = 0 if k. These ast two equaities impy that M is a right modue over H. Therefore, we have shown that the H- and the C[ S ]-modue structure on M can be gued to yied a modue over H. To prove that F is an equivaence, we are eft to show that it is fuy faithfu. That F is inective on morphisms is true because this is true for the Schur-Wey duaity functor between C[ S ] and ULs n, so suppose that f : FM 1 FM is a LY -homomorphism. From the main resuts of [ChPr1] and [Dr1], f is of the form fm v = gm v, m M 1, where g Hom C M 1, M is a inear map which is aso a homomorphism of right C[ S ]- and H-modues. Since H is generated by its two subagebras C[ S ] and H, g is even a homomorphism of H-modues. Therefore, f = Fg and this competes the proof of theorem Action of the eements X ± 0,1, H 0,1 Now that we know that FM is a modue over LY, it may be interesting to see expicity how the eements X 0,1 ± and H 0,1 act on it. What we wi discover wi be usefu in the next section. We wi assume throughout this section that λ = c, β = t nc c and n 3. 3

24 7.1 Action of X 0,1 Equations 51 and 5 yied X 0,1 = 1 [H 1,1 H n1,1, X 0 ] 1 λ βh1 βh n1 X 0 X 0 βh 1 λ βh n1. We wi use the notation K r z to denote the eement z u r Ls n for z s n ; in particuar, K 1 E n1 = X 0 and K 1E 1n = X0. The eement K rz maps to the operator in End C FM given by K r zm v = k=1 mxr k zk v. Writing H 1,1 as H 1,1 = JH 1 λν 1, and simiary for H n1,1, we can express X 0,1 in the foowing way. We wi use that [H n1 H 1, X 0 ] = 0. X 0,1 = 1 [JH 1 H n1, X 0 ] λ 8 n1 K1 E nd E d1 E d1 K 1 E nd d=3 λ 8 K1 E 11 K 1 E nn E n1 E n1 K1 E 11 K 1 E nn λ H 1X 0 X 0 H 1 λ K1 E n E 1 E 1 K 1 E n λ K1 E 1 E n E n K 1 E 1 8 λ n K1 E d1 E nd E nd K 1 E d1 λ K1 E 11 K 1 E nn E n1 8 8 d= E n1 K1 E 11 K 1 E nn λ K1 E n1,1 E n,n1 E n,n1 K 1 E n1,1 λ K1 E n,n1 E n1,1 E n1,1 K 1 E n,n1 8 λ X 0 H n1 H n1 X 0 1 λ βh1 βh n1 X 0 X 0 βh 1 λ βh n1 = 1 [JH 1 H n1, X 0 ] λ n K1 E nd E d1 E d1 K 1 E nd 8 d=3 λ K1 E n E 1 E 1 K 1 E n λ n K1 E d1 E nd E nd K 1 E d1 8 d=3 λ K1 E 11 K 1 E nn E n1 E n1 K1 E 11 K 1 E nn λ K1 E n1,1 E n,n1 E n,n1 K 1 E n1,1 = 1 [JE 11 E nn, X 0 ] 1 [H,1 H n,1, X 0 ] λ [ν ν n, X 0 ] 53 λ 8 n K1 E nd E d1 E d1 K 1 E nd λ n1 K1 E d1 E nd E nd K 1 E d1 8 d= λ K1 E n E 1 E 1 K 1 E n λ K1 H 0 E n1 E n1 K 1 H 0 8 λ K1 E n1,1 E n,n1 E n,n1 K 1 E n1,1 8 d=3

25 = 1 [JE nn E 11, X 0 ] λ 8 λ 8 n1 K1 E nd E d1 E d1 K 1 E nd d= n1 K1 E d1 E nd E nd K 1 E d1 λ K1 H 0 E n1 E n1 K 1 H 0 d= = JX 0 λ 8 ɛ [X 0, X ɛ ]X ɛ X ɛ [X 0, X ɛ ] λ 8 We define JX 0 to be 1 [JH 0, X 0 ]. Set Ỹ = 1 x Y Y x. JX 0 m v = 1 = λ = =1 =1 m[x k, Y ] H 0 Ek n1v 1 k=1 k K1 H 0 E n1 E n1 K 1 H 0. mx Y Y x E n1 v =1 mx k x s k H 0 Ek n1v k=1 k mỹ E n1 v =1 Ỹ E n1 λ X 8 0 H 0 H 0 X 0 E n1k 1 H 0 K 1 H 0 E n1 m v =1 Set JX 0 = JX 0 λ 8 X 0 H 0 H 0 X 0 E n1k 1 H 0 K 1 H 0 E n1, so X 0,1 = JX 0 λ [X 8 0, X ɛ ]Xɛ Xɛ [X 0, X ɛ ] λ 8 X 0 H 0 H 0 X 0 ɛ 7. Action of X 0,1 The action of X0,1 on FM can be expressed in a simpe way. Proceeding exacty as for X 0,1, we can write X0,1 = JX 0 λ [X 8 0, Xɛ ]X ɛ X ɛ [X0, X ɛ ] λ 8 X 0 H 0 H 0 X0 5 ɛ where JX0 acts on m v by JX 0 m v = 1 be written in the foowing form: =1 mx1 Y Y x 1 E 1n v. This can 1 =1 mx 1 Y Y x 1 E 1n v = 1 = =1 =1 m y 1 x1 m y 1 [x1 5 y x x y x 1 E 1n v, y ]x x [y, x 1 ] E 1n v

26 1 =1 mx 1 Y Y x 1 E 1n v = m y c =1 k = =1 m y c k x 1 s k k x 1 x 1 k s k E 1n v x 1 k s k E 1n v 55 As for the sum ɛ [X 0, X ɛ ]X ɛ X ɛ [X 0, X ɛ ], it equas n1 K1 E 1d E dn E dn K 1 E 1d K 1 E dn E 1d E 1d K 1 E dn K 1 H 0 E 1n E 1n K 1 H 0, d= so it acts on m v in the foowing way: ɛ [X 0, X ɛ ]X ɛ X ɛ [X 0, X ɛ ] m v = = = n1,k=1 k d=,k=1 k,k=1 k mx 1 k mx 1 k n d=1,k=1 k,k=1 k mx 1 k E k 1d E dn Ek dn E 1d v H k 0 E 1n v x 1 E1d k E dn v mx 1 k E k 1nE 11 Ek 1nE nnm v mx 1 x 1 k s k E 1n v X0 H 0 H 0 X0 m v 56 Combining equations 5,55 and 56, we concude that X0,1 m v = =1 my E 1n v. The eement X0,1 wi become important in the next section. We wi sometimes denote it by Y Action of H 0,1 We use the equaity H 0,1 = [X 0,0, X 0,1 ]. 6

27 H 0,1 m v = =1 = c = c = c = m[y, x k ] En1E k 1n v k=1 k =1 ms k En1E k 1n v k=1 k =1 k=1 k m E11E k nnv t c =1 =1 k=1 i=1 i s i my x Ennv =1 my x Ennv =1 my H 0 v =1 Ennv t c =1 m E11E k nnv JH 0 m v i=1 i s i t cn mx y E 11 v =1 mx y E 11 v =1 E 11 v E 11 E nn m v n n E nd E dn E dn E nd E 1d E d1 E d1 E 1d m v c E dd m v c d=1 JH 0 λ ɛ ɛ, θe ɛ E ɛ β λ E 11 E nn m v E ɛ E ɛ λ H 0 λ d=1 It can be proved that the subagebra of LY generated by Y 0 and s n is isomorphic to Us n C C[v]. See aso proposition 8.1 beow. We introduce the notation Q r z, r Z 0, to denote z v r as an eement of this subagebra; in particuar, Q 1 E 1n = Y 0. There are three types of operators in End C M C[S ] V which are of particuar interest to us: those coming from the action of Jz, K r z and of Q r z. They are reated to each other in the foowing way. Proposition 7.1 See aso [BHW]. Suppose that a b and c d. Then we have the equaity [Q 1 E ab, K 1 E cd ] [K 1 E ab, Q 1 E cd ] = δ bc JE ad δ da JE cb. Proof. First, we wi prove the equaity [Q 1 E 1n, K 1 H 0 ] [K 1 E 1n, Q 1 H 0 ] = JE 1n 57 [Q 1 E 1n, K 1 H 0 ] [K 1 E 1n, Q 1 H 0 ] = [ X0,1, [X 0, E 1n] ] 1 [ [E1n, [X 0, E 1n] ] ], [E n1, X0,1 ] = [H 0,1, E 1n ] 1 [H0, [X 0 [, E 1n] ] ], X0,1 1 [E1n, [X 0 [, H 0] ] ], X0,1 1 [E n1, [ E 1n, [[X 0, X 0,1 ], E 1n] ]] 7

28 = [H 0,1, E 1n ] [ [E 1n, X 0 ], ] 1 X 0,1 [E n1, [ E 1n, [H 0,1, E 1n ] ]] = [E 1n, H 0,1 ] 1 [ E n1, [ E 1n, [H 0,1, E 1n ] ]] 58 [H 0,1, E 1n ] = [ [H 0,1, E 1 ], E n ] [ E 1, [ E 3, [ [E n,n1, [H 0,1, E n1,n ] ] The expression = [X 1,1 βh 0X 1 λ βx 1 H 0, E n ] [ E 1, [ E 3, [ [E n,n1, X n1,1 λ βh 0X n1 βx n1 H 0 ] ] = [JX 1 λω 1 βh 0X 1 λ βx 1 H 0, E n ] [ E 1, [ E 3, [ [E n,n1, JX n1 λω n1 λ βh 0X n1 βx n1 H 0 ] ] = JE 1n βh 0 E 1n λ βe 1n H 0 βe n E 1 λ βe 1 E n λ[ω 1, E n] JE 1n λ[e 1,n1, ω n1 [ ] E 1, [ E 3, [ [E n,n1, λ βh 0 X n1 βx n1 H 0 ] ] [ E 1, [ E 3, [ [E n,n1, λ βh 0 X n1 βx n1 H 0 ] ] is equa to [E 1, λ βh 0 E n βe n H 0 ] = λ βe 1 E n βe n E 1 λ βh 0 E 1n βe 1n H n [E 1,n1, ω n1 ] = 1 E n E 1 E 1 E n 1 E1n E 11 E n1,n1 E 11 E n1,n1 E 1n = 1 E 1nH n1 H n1 E 1n 1 E n1,ne 1,n1 E 1,n1 E n1,n = 1 n1 E n E 1 E 1 E n 1 E 1nH 0 H 0 E 1n 61 = [ω 1, E n] = 1 n1 E n E 1 E 1 E n 1 E E nn E 1n E 1n E E nn =3 1 E 1nH 1 H 1 E 1n 1 E 1E n E n E 1 = 1 n1 E n E 1 E 1 E n 1 E 1nH 0 H 0 E 1n 6 = 8

29 Therefore, combining equations 59,61,6 and 60, we obtain the foowing simpe expression for [H 0,1, E 1n ]: [H 0,1, E 1n ] = JE 1n λ H 0E 1n E 1n H 0 63 Putting together equations 58 and 63 yieds equaity 57: [QE 1n, K 1 H 0 ] [K 1 E 1n, QH 0 ] = JE 1n λh 0 E 1n E 1n H 0 1 [E n1, [E 1n, JE 1n λ H 0E 1n E 1n H 0 ]] = JE 1n λh 0 E 1n E 1n H 0 λ[e n1, E1n] = JE 1n The bracket of E n1 with both sides of equation 57 yieds [K 1 E n1, Q 1 E 1n ] [Q 1 E n1, K 1 E 1n ] = JH 0 6 This proves proposition 7.1 when a = n, b = 1, c = 1, d = n. Assuming that a 1, n, we appy [E an, ] to 6 to get [K 1 E a1, Q 1 E 1n ] [Q 1 E a1, K 1 E 1n ] = JE an. If b 1, a, we appy [, E 1b ] to the previous equation: this yieds [K 1 E ab, Q 1 E 1n ] [Q 1 E ab, K 1 E 1n ] = 0. If c 1, n, we use [E c1, ] to get [K 1 E ab, Q 1 E cn ] [Q 1 E ab, K 1 E cn ] = δ bc [K1 E a1, Q 1 E 1n ] [Q 1 E a1, K 1 E 1n ] = δ bc JE an We now appy [, E nd ] if b, d n and obtain [K 1 E ab, Q 1 E cd ] [Q 1 E ab, K 1 E cd ] δ ad [K1 E nb, Q 1 E cn ] [Q 1 E nb, K 1 E cn ] = δ bc JE ad δ bc δ ad JE nn. Note that, athough JE nn is not defined, if b = c and a = d, then the right-hand side becomes JE aa E nn. It is enough to show that [K 1 E nb, Q 1 E cn ] [Q 1 E nb, K 1 E cn ] = JE cb δ bc JE nn. Starting with 6 and assuming that b, c 1, n, we appy [, E 1b ] and [E c1, ] to get this ast equation. The remaining cases can be handed in a simiar manner. 8 Schur-Wey dua of the rationa Cherednik agebra Our goa in this section is to estabish an equivaence of categories for the rationa Cherednik agebra simiar to the one given in theorem 5. and to identify the Schur-Wey dua of H with a subagebra of LY. 9