arXiv:q-alg/9710031v2 21 Apr 1998 Degenerate Double Affine Hecke Algebra and Conformal Field Theory Tomoyuki Arakawa, Takeshi Suzuki†and Akihiro Tsuchiya Abstract We introduce a class of induced representations of the degenerate double affine Hecke algebra H of glN (C) and analyze their structure mainly by means of intertwiners of H . We also conb (C)-modules using Knizhnik-Zamolodchikov struct them from sl m connections in the conformal field theory. This construction provides natural quotients of the induced modules, which correspond b (C)-modules. Some conjectural formulas are to the integrable sl m presented for the symmetric part of them. Introduction In this paper, the representations of the degenerate double affine Hecke algebra H are discussed from view points of the conformal field theory b (C). The relations between the associated to the affine Lie algebra sl m KZ-connections of the conformal field theory and the representations of the degenerate affine Hecke algebra was first discussed by Cherednik [1]. And Matsuo [2] succeeded to clarify the relations between the differential equations satisfied by spherical functions and KZ-connections. At first part of this paper we discuss the properties of the parabolic induced modules of H, which are induced from a certain one dimensional representations of parabolic subalgebras of H. Secondly we will give an explicit construction of H-modules from b (C) modules through KZ-connections. It will be shown that these Hsl m b (C) correspond to parabolic modules arising from Verma modules of sl m induced modules of H. In the final part of this paper, we will discuss the structure of the representations of affine Hecke algebra arising from level ℓ integrable repb (C). resentations of sl m † Supported by JSPS the Research Fellowships for Young Scientists. 1 DDAHA & CFT 2 We describe the contents of the paper more precisely: In §1 we introduce basic notions about the degenerate double affine Hecke algebra H. In particular, intertwiners of weight spaces of H play essential roles in the analysis of H-modules. In §2, we introduce parabolic induced modules and investigate their structure when the parameter is generic (Definition 2.4.1), and the results are these (Proposition 2.4.3, Theorem 2.4.6, Corollary 2.4.7): (1) The irreducibility of the standard modules are shown and their basis are described by intertwining operators. (2) Decompositions of the standard modules as H̄-modules are obtained. (3) The symmetric part of the standard modules are decomposed into weight spaces with respect to the action of the center of H̄, and their basis are constructed again by using intertwiners. In non-generic case, we present a sufficient condition for an induced module to have a unique irreducible quotient (Corollary 2.5.4). §3 is devoted to some preliminaries on affine Lie algebras and in §4, we b (C)realize H-modules as a quotient space of a tensor product of g = sl m modules. More precisely, for g-modules A, B, we consider the space F (A,     .   B)  ±1 ±1 m m ⊗B , ⊗ B g′ A ⊗ ⊗N = A ⊗ ⊗N i=1 C[zi ] ⊗ C i=1 C[zi ] ⊗ C where C[zi±1 ] ⊗ Cm is an evaluation module of g and g′ = [g, g] acts diagonally on the tensor product. By combining the Knizhnik-Zamolodchikov connection with the Cherednik-Dunkl operator, we define an action of H on F (A, B) (Theorem 4.2.2). In §5, we construct the isomorphism between a parabolic induced module and M(µ, λ) := F (M(µ), M ∗ (λ)) for highest and lowest Verma module M(µ) and M ∗ (λ), λ and µ being weights of g (Proposition 5.2.3). Since our construction turns out to be functorial, V(µ, λ) := F (L(µ), L∗ (λ)) gives a quotient module of M(µ, λ), where L(µ) and L∗ (λ) are the irreducible quotients of M(µ) and M ∗ (λ) respectively. We focus on the case where λ and µ are both dominant integral weights and study about V(µ, λ). In the last section §6, we focus on the symmetric part of V(µ, λ) for dominant integral weights λ, µ, and present a description of the basis by intertwiners (Conjecture 6.2.6) as a consequence of the character formula (85), which is still conjectural since it is proved under the assumption that a certain sequence of H-modules (coming from BGG exact sequence of g) is exact (Conjecture 6.1.1). It is interesting that character formula (85) also appears in the theory DDAHA & CFT 3 of the solvable lattice model [10]. Preliminaries In this section, we review the basic notions about the degenerate double affine Hecke algebra H. 1.1 Affine root system L ∨ Let t̄ = N i=1 Cǫi be the Cartan subalgebra of glN (C) with the invariant bilinear form (ǫ∨i , ǫ∨j ) = δij . Define the Cartan subalgebra t of the affine Lie b (C) by t = t̄ ⊕ Cc ⊕ Cd. Extend the non-degenerate invariant algebra gl N symmetric bilinear form (, ) to t by putting (c, d) = 1 and (ǫi , c) = (ǫi , d) = L ∗ (c, c) = (d, d) = 0. Let t̄∗ = N i=1 Cǫi be the dual space of t̄ and t = t̄∗ ⊕ Cδ ⊕ Cc∗ be the dual space of t, where ǫi , δ and c∗ are the dual vectors of ǫ∨i , d and c respectively. We often identify t∗ with t via the correspondences ǫi 7→ ǫ∨i , δ 7→ c and c∗ 7→ d. Let ζ ∨ ∈ t denote the vector corresponding to ζ ∈ t∗ . Define the systems R of roots, R+ of positive roots and Π of simple (1) roots of type AN −1 by n o R = α + kδ | α ∈ R̄, k ∈ Z , n o n o R+ = α + kδ | α ∈ R̄+ , k ≥ 0 ⊔ −α + kδ | α ∈ R̄+ , k > 0 , Π = {α0 := δ − (ǫ1 − ǫN )} ⊔ Π̄, where R̄, R̄+ and Π̄ are the systems of roots, positive roots and simple roots of type AN −1 respectively: R̄ = {αij = ǫi − ǫj | i 6= j} , R̄+ = {αij | i < j} , Π̄ = {α1 , . . . , αN −1 } (αi = αii+1 ). 1.2 Affine Weyl group Let t′ = t̄ ⊕ Cc ⊂ t. We consider the dual space (t′ )∗ of t′ as a subspace of t∗ via the identification (t′ )∗ = t∗ /Cδ ∼ = t̄∗ ⊕ Cc∗ . LN −1 Let Q̄ be the root lattice i=1 Zαi and L P̄ be the weight lattice N i=1 Zǫi . Let W̄ be the Weyl group of glN (C), which is isomorphic to the symmetric group SN . The affine Weyl group W is defined as a semidirect product W = W̄ ⋉ P̄ , DDAHA & CFT 4 with the relation w · tη · w −1 = tw(η) , where w and tη are the elements in W corresponding to w ∈ W̄ and η ∈ P̄ respectively. The group W contains (1) the affine Weyl group W a = W̄ ⋉ Q̄ of type AN −1 as its subgroup. Let sα ∈ W̄ be the reflection corresponding to α ∈ R̄. The action of W on an element ξ ∈ t is given by the following formulas: sα (ξ) = ξ − α(ξ)α∨    α ∈ R̄ ,  tη (ξ) = ξ + δ(ξ)η ∨ − η(ξ) + 21 (η, η)2 δ(ξ) c The dual action on t∗ is given by sα (ζ) = ζ − (α, ζ)α    α ∈ R̄, ζ ∈ t∗ ,  tη (ζ) = ζ + (δ, ζ)η − (η, ζ) + 21 (η, η)2 (δ, ζ) δ    (1) η ∈ P̄ .  η ∈ P̄ , ζ ∈ t∗ . (2) With respect to these actions, the inner products in t and t∗ are W invariant. The action (2) preserves the set R of roots. The following action on (t′ )∗ is called the affine action:   sα (ζ) = ζ − (α, ζ)α α ∈ R̄, ζ ∈ (t′ )∗ ,   tη (ζ) = ζ + (δ, ζ)η η ∈ P̄ , ζ ∈ (t′ )∗ . (3) For an affine root α = ᾱ + kδ (ᾱ ∈ R̄, k ∈ Z), define the corresponding affine reflection by sα = t−kᾱ ·sᾱ . Set si = sαi for i = 0, . . . , N −1. We often identify the set {0, . . . , N − 1} with the abelian group Z/NZ throughout this article. Let π = tǫ1 · s1 · · · sN −1 . The following is well-known. Proposition 1.2.1 The group W is isomorphic to the group defined by the following generators and relations: generators : si (i ∈ Z/NZ), π ±1 , relations : s2i = 1 (i ∈ Z/NZ), si · si+1 · si = si+1 · si · si+1 (i ∈ Z/NZ), si · sj = sj · si (i − j 6≡ ±1 mod N), π · si = si+1 · π (i ∈ Z/NZ), π · π −1 = 1, and the subgroup W a is generated by the simple reflections s0 , . . . , sN −1 . In particular, W ∼ = Ω ⋉ W a, where Ω = hπ ±1 i ∼ = Z. For w ∈ W , let S(w) = R+ ∩ w −1 (R− ), where R− is the set of negative roots R\R+ . The length l(w) of w ∈ W is defined as the number ♯S(w) of the elements in S(w). For w ∈ W , an expression w = π k · sj1 · · · sjm is called a reduced expression if m = l(w). Put S̄(w) = S(w) ∩ R̄+ . DDAHA & CFT 5 Remark 1.2.2 The elements of the abelian subgroup Ω are characterized as elements of length 0 in W , i.e., Ω = {w ∈ W | l(w) = 0}. The partial ordering  called the Bruhat ordering is defined in the Coxeter group W a : w  w ′ if w can be obtained as a subexpression of a reduced expression of w ′ . Extend this ordering  to the partial ordering in W by ′ π k w  π k w ′ ⇔ k = k ′ and w  w ′ (w, w ′ ∈ W a ). 1.3 Degenerate double affine Hecke algebra Let C[W ] denote the group algebra of W and S[t′ ] denote the symmetric algebra of t′ . Clearly C[W ] = C[P̄ ] ⊗ C[W̄ ] and S[t′ ] = S[t̄′ ] ⊗ C[c]. The degenerate double affine Hecke algebra was introduced by Cherednik ([1]). Definition 1.3.1 The degenerate double affine Hecke algebra (DDAHA) H is the unital associative C-algebra defined by the following properties: (i) As a C-vector space, H∼ = C[W ] ⊗ S[t′ ]. (ii) The natural inclusions C[W ] ֒→ H and S[t′ ] ֒→ H are algebra homomorphisms (the images of w ∈ W and ξ ∈ t̄′ will be simply denoted by w and ξ). (iii) The following relations hold in H: si · ξ − si (ξ) · si = −αi (ξ) (i = 0, . . . , N − 1, ξ ∈ t′ ), π · ξ = π(ξ) · π (ξ ∈ t′ ). (4) (5) Remark 1.3.2 By definition, the element c ∈ H belongs to the center Z(H). And the original algebra defined in [1] is the quotient algebra H(κ) = H/hc − κ · id i (κ ∈ C∗ ). Definition 1.3.3 Define the degenerate affine Hecke algebra (DAHA) H̄ as the following subalgebra of H: H̄ = hw ∈ W̄ , ξ¯ ∈ t̄i ∼ = C[W̄ ] ⊗ S[t̄]. The following proposition is easy to prove (see [3]). DDAHA & CFT 6 Proposition 1.3.4 (i) For w ∈ W and ξ ∈ t′ , we have  In particular, ξ · w = w ·w −1(ξ) + X α∈S(w) ξ · w = w · w −1 (ξ) +  w(α)(ξ)sα . X cw ′ w ′ , w ′ ≺w for some cw′ ∈ C. (ii) For p ∈ S[t′ ] and i = 0, . . . , N − 1, we have p · si − si · si (p) = −△i (p), where △i (p) = 1 (p α∨ i (6) − si (p)) ∈ S[t′ ]. (w ′ ∈ W a ). Note that H = S[t′ ] · C[W ] = C[W ] · S[t′ ] from (i) of the above proposition. Remark 1.3.5 Let S(t′ ) be the quotient field of S[t′ ]. Then one can natf = H ⊗ ′ S(t′ ) ∼ urally extend the definition of H to the algebra H = S[t ] C[W ] ⊗ S(t) by using Eq. (6) in place of Eq. (4) Proposition 1.3.6 (i) The center Z(H) of H equals C[c]. (ii) The center Z(H̄) of H̄ equals the W̄ -invariant part S[t̄]W̄ of S[t̄]. Outline of Proof.) (cf. [4]) (i) Let ZS[t′] (H) denote the elements in H which commutes with S[t′ ]. By the above Proposition 1.3.4 (i), it follows that ZS[t′] (H) = S[t′ ]. Then by (ii) of the proposition, one can see that Z(H) = S[t′ ]W = C[c]. The proof for (ii) is the same.  Let us identify the group algebra C[P̄ ] with the Laurent polynomial ± ring C[z1± , . . . , zN ] by putting zi = eǫi ∈ C[P̄ ]. The following proposition gives another description of the algebra H: Proposition 1.3.7 The algebra H is the unital associative C-algebra such that H = C[P̄ ] ⊗ H̄ ⊗ C[c] as a C-vector space, c ∈ Z(H),   w · f · w −1 = w(f ) w ∈ W̄ , f ∈ C[P̄ ] , h i   ) ¯ (1−sᾱ−)(f ¯ f = c ∂ξ̄ (f ) + Pᾱ∈R̄ ᾱ(ξ) ¯ ∈ t̄, f ∈ C[P̄ ] , ξ, · s ξ ᾱ ᾱ + 1−e ¯ η̄ (ξ ∈ t̄, η̄ ∈ P̄ ) and the natural inclusions C[P̄ ] ֒→ H, where ∂ξ̄ ·eη̄ = η̄(ξ)e H̄ ֒→ H and C[c] ֒→ H are algebra homomorphisms. DDAHA & CFT 7 Proof. follows directly from Definition 1.3.1 and Proposition 1.3.4.  ≥0 Let W ⊂ W be the subsemigroup generated by si (i = 1, . . . , N) and π. Define H≥0 = C[W ≥0 ] ⊗ S[t′ ] ∼ = C[P̄ ≥0] ⊗ H̄ ⊗ C[c], L ≥0 where P̄ ≥0 = N is the subalgebra i=1 Z≥0 ǫi . Then it is easy to see that H of H by the above proposition. 1.4 Intertwiners For an H-module V and ζ ∈ (t′ )∗ , define the weight space Vζ and the generalized weight space Vζgen with respect to the action of S[t′ ]: Vζ = {vn∈ V | ξ · v = ζ(ξ)v for any ξ ∈ t′ } o Vζgen = v ∈ V | (ξ − ζ(ξ))k · v = 0 for any ξ ∈ t′ , for some k ∈ N . Let ϕi = 1 + si αi∨ ∈ H (i ∈ {0, . . . , N − 1} ∼ = Z/NZ), ϕπ = π ∈ H. Then ϕi · ξ = si (ξ) · ϕi , ϕπ · ξ = π(ξ) · ϕπ , (7) (8) for ξ ∈ t′ and i ∈ Z/NZ (Eq. (8) is nothing but the defining relation (5)). Proposition 1.4.1 [4] The above defined elements satisfy the following relations: ϕi · ϕi+1 · ϕi = ϕi+1 · ϕi · ϕi+1 (i ∈ Z/NZ), ϕi · ϕj = ϕj · ϕj (i − j 6= ±1 mod N) , ϕπ · ϕi = ϕi+1 · ϕπ (i ∈ Z/NZ), ϕ2i = 1 − αi∨ 2 (i ∈ Z/NZ). Proof. We only show ϕi · ϕi+1 · ϕi = ϕi+1 · ϕi · ϕi+1 . The rest is easy. Note f ¿From Proposition 1.3.4 (i), one can prove that that ϕi has its inverse in H. n o f = S(t′ )·C[W ] and Z ′ (H) f := X ∈ H f | [X, p] = 0 for all p ∈ S(t′ ) = H S(t ) f by Eq. (7), it S(t′ ). Since (ϕi · ϕi+1 · ϕi ) · (ϕi+1 · ϕi · ϕi+1 )−1 ∈ ZS(t′ ) (H) follows that ϕi · ϕi+1 · ϕi = p · ϕi+1 · ϕi · ϕi+1 for some p ∈ S(t′ ). By comparing the coefficient of si ·si+1 ·si = si+1 ·si ·si+1 of both sides, we get the desired equality.  DDAHA & CFT 8 Let ϕw = ϕkπ · ϕj1 · · · ϕjl ∈ H, for w = π k · sj1 · · · sjl ∈ W (reduced expression). These elements are well-defined by the Proposition 1.4.1, and Eq.(7) reads as ϕw · ξ = w(ξ) · ϕw (w ∈ W, ξ ∈ t′ ) . (9) Hence, we have Proposition 1.4.2 Let V be an H-module. Let ζ ∈ (t′ )∗ and w ∈ W . Then ϕw · v ∈ Vw(ζ) for any v ∈ Vζ . Here recall that the action of W on (t′ )∗ is the affine action (defined by Eq (3)). The element ϕw is called the intertwiner (of weight spaces). Proposition 1.4.3 Let w ∈ W . (i) Y X ϕw = w · α∨ + x · fx , x≺w α∈S(w) for some fx ∈ S[t′ ]. (ii) ϕw−1 · ϕw = Y 2 (1 − α∨ ). α∈S(w) Proof. (i) follows from the well-known fact S(w) = {sjl · · · sj2 (αj1 ), sjl · · · sj3 (αj2 ), . . . , αjl } for w = π k sj1 sj2 · · · sjl (reduced expression). (ii) follows from the last relation of Proposition 1.4.1.  Lemma 1.4.4 If w ∈ W ≥0 , then ϕw ∈ H≥0 . Representations In this section we study some important class of representations of H for the next section. 2.1 Induced representations Recall that every irreducible representation of the degenerate affine Hecke algebra H̄ can be obtained as the unique irreducible quotient of a standard module (see [5]), which is a representation induced from some parabolic subalgebra” of H̄. Hence it is natural to start with investing such induced modules: DDAHA & CFT 9 Let us denote β ⊢ N if an ordered sequence β = (β1 , . . . , βm ) of positive P integers is a (ordered) partition of N, i.e., m i=1 βi = N. (1) (2) (m) For a given β = (β1 , . . . , βr ) ⊢ N, let Iβ = (Iβ , Iβ , . . . , Iβ ), o nP Pk−1 Pk (k) k−1 where Iβ = Define R̄β = i=1 βi + 1, i=1 βi + 2, . . . , i=1 βi . n o αij ∈ R̄ | i, j ∈ Iβk for some k , R̄β.+ = R̄+ ∩ R̄β and Π̄β = Π̄ ∩ R̄β . Let W̄β be the subgroup of W̄ generated by sα (α ∈ Π̄β ). Then clearly W̄β is a parabolic subgroup of W̄ : W̄β ∼ = W̄β1 × · · · × W̄βr (W̄βj = Sβj ). (10) Define subalgebras Hβ = S[t′ ] ⊗ C[W̄β ] ⊂ H, H̄β = S[t̄] ⊗ C[W̄β ] ⊂ H̄. (11) We call Hβ (resp. H̄β ) the parabolic subalgebra of H (resp. of H̄) associated with β ⊢ N. Clearly Hβ = H̄β ⊗ C[c]. Let (t′ )∗β = t̄∗β = n o ζ ∈ (t′ )∗ | ζ(α) = −1 for α ∈ Π̄β ⊂ (t′ )∗ , (12) ζ ∈ t̄∗ | ζ(α) = −1 for α ∈ Π̄β ⊂ t̄∗ . (13) n o Then an element ζ ∈ (t′ )∗β (resp. ζ ∈ t̄∗β ) defines a well-defined onedimensional representation C1ζ of Hβ (resp. of H̄β ): w · 1ζ = 1ζ for all w ∈ W̄β , ξ · 1ζ = ζ(ξ)1ζ for all ξ ∈ t′ (resp. ξ ∈ t̄ ). Definition 2.1.1 Define the induced representation Yβ (ζ) of H and Ȳβ (ζ) of H̄ by Yβ (ζ) = H ⊗Hβ C1ζ , Ȳβ (ζ) = H̄ ⊗H̄β C1ζ . The cyclic vectors 1 ⊗ 1ζ will be denoted by 1ζ . Clearly, Yβ (ζ) ∼ = C[W/W̄β ] as W -module, Ȳβ (ζ) ∼ = C[W̄ /W̄β ] as W̄ -module. Let ζ̄ be the image of ζ ∈ (t′ )∗β by the projection (t′ )∗β → t̄∗β . Then, Yβ (ζ) ∼ = C[P̄ ] ⊗ Ȳβ (ζ̄), = H(κ) ⊗H̄ Ȳβ (ζ̄) ∼ where H(κ) = H/hc − κ idi and κ = ζ(c). DDAHA & CFT 2.2 10 Basis Define the following subsets of W for β ⊢ N: n o W β = w ∈ W | l(w · u) ≥ l(w) for any u ∈ W̄β , W̄ β = W β ∩ W̄ . The following well-known proposition will be frequently used in the rest of this section. Proposition 2.2.1 n o n o β (i) W = w ∈ W | S(w) ⊂ R+ \R̄β,+ and W̄ β = w ∈ W̄ | S(w) ⊂ R̄+ \R̄β,+ . (ii) For any w ∈ W (resp. W̄ ), there exist a unique w1 ∈ W β (resp. W̄ β ) and a unique u ∈ W̄β , such that w = w1 · u. Their length satisfy l(w) = l(w1 ) + l(u). In particular, the set W β (resp. W̄ β ) gives a complete representatives in the coset W/W̄β (resp. W̄ /W̄ β ). o n Hence the space Yβ (ζ) has a basis w · 1ζ | w ∈ W β with the partial ordering  among them induced from the Bruhat ordering. Let us give more precise description of this basis: Definition and Proposition 2.2.2 For η ∈ P̄ , define γη as the element of W̄ with shortest length possible such that γη (η) ∈ P̄− . Then, S(γη ) = n o α ∈ R̄+ | (η, α) > 0 . Example 2.2.3 Let η = aǫ1 + b(ǫ2 + ǫ3 ) + c(ǫ4 + ǫ5 + ǫ6 )!+ b(ǫ7 + ǫ8 + ǫ9 ) 1 2 3 4 5 6 7 8 9 . with b < c < a. Then γη = 9 1 2 6 7 8 3 4 5 Lemma 2.2.4 (i) For the partition (N) (the partition such that I(N ) = {1, 2, . . . , N}), n o W (N ) = tη · γη−1 | η ∈ P̄ . (ii) For a general partition β ⊢ N, W β = W (N ) · W̄ β . Moreover, l(w) = l(tη · γη−1 ) + l(x) for w = tη · γη−1 · x (η ∈ P̄ , x ∈ W̄ β ). Proof. follows from Proposition 2.2.1 and Definition and Proposition 2.2.2.  DDAHA & CFT 2.3 11 Representatives in double coset In this subsection we want to give a description of the double coset W̄ \W/W̄β , which is related to the decomposition of Yβ (ζ) into a direct sum of H̄-modules. n In this subsection β ⊢ N is fixed. o Let P̄− = η ∈ P̄ | (η, α) ≤ 0 for any α ∈ R̄+ and P̄β,− = n o η ∈ P̄ | (η, α) ≤ 0 for any α ∈ R̄β,+ . It is not difficult to see the following lemma: Lemma 2.3.1 The set P̄β,− gives complete representatives in the double coset W̄ \W/W̄β . For later purposes, let us give another description of the representatives in the double coset: Definition 2.3.2 For each w ∈ W̄ β , define the element ηw ∈ P̄− by the following conditions: (ηw , ǫ1 ) = 0, (ηw , αi ) = ( Set −1 0 if αi ∈ w(R̄+ \R̄β,+ ) otherwise n (i = 1, . . . , N−1). o P̄− (w) = η + ηw | η ∈ P̄− ⊂ P̄− , n o Xβ = tη · w | w ∈ W̄ β , η ∈ P̄− (w) . Note that γη = 1 for η ∈ P̄− and thus X β ⊆ W β by Lemma 2.2.4. n o Lemma 2.3.3 P̄− (w) = η ∈ P̄− | (η, α) < 0 for α ∈ w(R̄+ \R̄β,+ ) ∩ R̄+ . Proof. It is easy to see that the right hand side is included in P̄− (w). To prove the opposite inclusion, we suppose that η ∈ P̄− (w) and (η, αij ) = 0 (i < j) and will show that αij ∈ / w(R̄+ \R̄β,+ ) ∩ R̄+ . Since η ∈ P̄− , it follows that (η, αij ) = 0 implies (η, αk ) = 0, and thus αk ∈ / w(R̄+ \R̄β,+ ) for / w(R̄+ \R̄β,+ ) ∩ R̄+ follows using R̄+ = w(R̄β,+ ) ⊔  i ≤ k < j. Now αij ∈ w(R̄+ \R̄β,+ ) ∩ R̄+ ⊔ S(w −1 ). Definition 2.3.4 Let η ∈ P̄β,− . We define η− : the unique element of P̄− ∩ {w(η) | w ∈ W̄ }, ω̃η : the unique longest element in W̄ such that ω̃η (η) = η− , ωη : the unique shortest element in ω̃η W̄β . The following lemma is easy to see. Lemma 2.3.5 Let η ∈ P̄β,− and let w ∈ W̄ be such that w(η) = η− . Suppose that si ∈ W̄β and l(wsi ) < l(w). Then si (η) = η. DDAHA & CFT 12 Lemma 2.3.6 For η ∈ P̄β,− , we have (i) η− ∈ P̄− (ωη ). (ii) ωη (η) = η− . (iii) W̄ tη W̄β = W̄ tη− ωη W̄β . Proof. Put ω = ωη . To prove (i), we shall show (η− − ηω , αi ) ≤ 0 for i = 1, . . . , N − 1. This holds obviously if (η− , αi ) < 0 or (ηω , αi ) = 0. We suppose that (η− , αi ) = 0 and αi ∈ ω(R̄+ ), and we shall show αi ∈ ω(R̄β,+ ). Note that this implies (η− , αi ) = 0 ⇒ (ηω , αi ) = 0 and thus completes the proof of η− ∈ P̄− (ω). Write ω = ω̃z, where ω̃ = ω̃η and z ∈ W̄β . Since (η− , αi ) = 0 we have si ω̃(η) = si (η− ) = η− . Hence l(si ω̃) < l(ω̃), and this is equivalent to ω̃ −1 (αi ) ∈ R̄− . Combining with αi ∈ ω(R̄+ ), we have ω −1 (αi ) ∈ R̄+ ∩ z −1 (R̄− ), which is a subset of R̄β,+ as z ∈ W̄β . Let us prove (ii). Since ω(η) = ω̃z(η), it is enough to prove z(η) = η. If z = si1 si2 · · · sip is a reduced expression of z, then we have si1 , si2 , . . . , sip ∈ W̄β and l(ω̃) > l(ω̃si1 ) > l(ω̃si1 si2 ) > · · · > l(ω̃si1 · · · sip ) = l(ω̃z). Now the statement follows from Lemma 2.3.5. The statement (iii) is a direct consequence of (ii).  Proposition 2.3.7 The set X β ⊆ W gives complete representatives in the double coset W̄ \W/W̄β . Proof. By Lemma 2.3.6, we can define the map f : P̄β,− → X β by f (η) = tη− ωη (η ∈ P̄β,− ). We define the map g : X β → P̄ by g(tη w) = w −1(η) (w ∈ W̄ β , η ∈ P̄− (w)). Let α ∈ R̄β,+ . Then w(α) ∈ R̄+ by Proposition 2.2.1-(i). Therefore (w −1(η), α) = (η, w(α)) ≤ 0 and thus g(tη w) ∈ P̄β,− . Namely the image of g is included in P̄β,− . By Lemma 2.3.6-(ii), we have g ◦ f = idP̄β,− . Let us check f ◦ g = idX β . For tη′ w ∈ X β , put η = w −1 (η ′) ∈ P̄β,− . Then f ◦ g(tη′ w) = tη− ωη . It is easy to see η ′ = η− . We will show w = ωη . Since η ′ ∈ P̄− (w), it can be shown using Lemma 2.3.3 that S(w) = {α ∈ R̄+ \ R̄β,+ | (η, α) ≥ 0}. Similarly, by η− = η ′ ∈ P̄− (ωη ), we have S(ωη ) = {α ∈ R̄+ \ R̄β,+ | (η, α) ≥ 0} = S(w). (14) Thus we have w = ωη as required. Now the statement follows from Lemma 2.3.1 and Lemma 2.3.6-(iii).  DDAHA & CFT 13 Remark 2.3.8 Let η ∈ P̄β,− . The formula (14) implies that ωη is also characterized as the longest element such that ωη ∈ W̄ β and ωη (η) = η− . (15) Definition 2.3.9 For w ∈ W , define β(w) as an ordered partition of N such that Π̄β(w) = Π̄ ∩ w(R̄β,+ ). Set W̄β(w) = hsα | α ∈ Π̄β(w) i, W̄[w] = {u ∈ W̄ | uw W̄β = w W̄β }. The proof of the following lemma is elementary. Lemma 2.3.10 (i) For w ∈ W β , the followings are equivalent: (a) si w ∈ / W β . (b) si ∈ W̄β(w) . (c) si ∈ W̄[w] . (ii) Let x = tη w ∈ X β (η ∈ P̄− (w), w ∈ W̄ β ). Then W̄[x] = W̄η ∩ W̄[w] , where W̄η = {u ∈ W̄ | w(η) = η}. Proposition 2.3.11 For x ∈ X β , we have W̄[x] = W̄β(x) . Proof. Let x = tη w ∈ X β (η ∈ P̄− (w), w ∈ W̄ β ). It is obvious that W̄β(x) ⊆ W̄[x] . We will prove u ∈ W̄[x] ⇒ u ∈ W̄β(x) by the induction on l(u). The case l(u) = 1 has been proved in Lemma 2.3.10-(i). Let l(u) > 1. It is enough to show that there exists si ∈ W̄[x] such that si u ∈ W̄[x] and l(u) = l(si u) + 1. Because W̄η is generated by the simple refrections si such that (η, αi ) = 0, we can find si ∈ W̄η such that l(u) = l(si u) + 1. We will show si ∈ W̄[x] , or equivalently, v := si u ∈ W̄[x] . If si x ∈ / W β , then si ∈ W̄[x] by Lemma 2.3.10-(i). Suppose si x ∈ W β . Then, noting R̄+ = w(R̄β,+ ) ⊔ (w(R̄+ \ R̄β,+ ) ∩ R̄+ ) ⊔ S(w −1 ), (16) we have si ∈ S(w −1 ). It follows from u = si v ∈ W̄[x] ⊆ W̄[w] that S(w −1 ) ⊆ S(w −1v −1 si ) and in particular, w −1 v −1 si (αi ) = −w −1 v −1 (αi ) ∈ R̄− . Hence v −1 (αi ) ∈ / S(w −1 ). Since (η, v −1(αi )) = (v(η), αi ) = 0, we have v −1 (αi ) ∈ w(R̄β,+ ) (by using (16) again). Hence w W̄β = uw W̄β = si vw W̄β = vsv−1 (αi ) w W̄β = vw W̄β . Therefore v ∈ W̄[x] by Lemma 2.3.10(ii).  DDAHA & CFT 14 Corollary 2.3.12 Let x ∈ X β . We have W̄ β(x) x ⊆ W β . Moreover, the map G W̄ β(x) → W β x∈X β given by w 7→ wx (w ∈ W̄ β(x) ) is bijective. 2.4 Structure in generic In this section β ⊢ N is still fixed. Definition 2.4.1 A weight ζ ∈ (t′ )∗β (resp. t̄∗β ) is said to be β-generic if (ζ, α) 6∈ {1, 0, −1} for any α ∈ ∪w∈W β S(w) (resp. α ∈ ∪w∈W̄ β S̄(w)). Remark 2.4.2 A weight ζ ∈ (t′ )∗β (resp. t̄∗β ) is β-generic if and only if (ζ, α) 6∈ {1, 0, −1} for any α ∈ R+ \R̄β,+ (resp. R̄+ \R̄β,+ ). Proposition 2.4.3 If ζ ∈ (t′ )∗β (resp. t̄∗β ) is β-generic, then Yβ (ζ) ( resp. Ȳβ (ζ)) is ano irreducible H̄-module) with a basis o n H-module ( resp. n β β (resp. ϕw · 1ζ | w ∈ W̄ ). ϕ w · 1ζ | w ∈ W Proof. Let ζ ∈ (t′ )∗β . By Proposition 1.4.3 (i) and Proposition 2.2.1 (ii), P each vector ϕw · 1ζ has the form of ϕw 1nζ = c · w1ζ + v≺w o cv v1ζ for some β c 6= 0 and cv ∈ C. Hence the elements ϕw · 1ζ | w ∈ W form a basis of Yβ (ζ). Moreover, by Proposition 1.4.3 (ii), each ϕw (w ∈ W β ) is invertible on the vector 1ζ . The proof of the statement about Ȳβ (ζ) is the same.  Any weight ζ ∈ (t′ )∗ defines a character [ζ] : S[t′ ]W̄ = S[t̄]W̄ ⊗C[c] → C via the evaluation at ζ. It is regarded as an element of (t′ )∗ /W̄ . (Recall that S[t̄]W̄ is the center of H̄.) Proposition 2.4.4 Suppose that ζ ∈ (t′ )∗β (resp. t̄∗β ) is β-generic and (ζ, δ) 6= 0. Then (i) The map W → (t′ )∗ (resp W̄ → t̄∗ ) given by w 7→ w(ζ) is injective. (ii) The map X β → (t′ )∗ /W̄ given by x 7→ [x(ζ)] is injective. Proof. The β-genericity of ζ ∈ (t′ )∗β implies that (ζ, α) 6= 0 for any α ∈ R. ¿From this, the statements follow easily.  Lemma 2.4.5 Let ζ ∈ (t′ )∗β . Let w ∈ W β and αi ∈ Π̄β(w) . Then si · ϕw · 1ζ = ϕw · 1ζ for the cyclic vector 1ζ ∈ Yβ (ζ)ζ . DDAHA & CFT 15 Proof. Since si w = wsw−1(αi ) and w −1 (αi ) ∈ R̄β,+ , we have l(wsw−1(αi ) ) = l(si w) = l(w) + 1. By Proposition 2.2.1-(ii), we have l(wsw−1(αi ) ) = l(w) + l(sw−1 (αi ) ). Hence ϕi · ϕw · 1ζ = ϕsiw · 1ζ = ϕwsw−1 (α ) · 1ζ = ϕw · ϕsw−1 (α ) · 1ζ . i i Noting (w(ζ), αi ) = −1 and sw−1 (αi ) 1ζ = 1ζ , we have (1 − si )ϕw 1ζ = ϕi · ϕw 1ζ = ϕw · ϕsw−1 (α ) 1ζ = ϕw · (1 − sw−1 (αi ) )1ζ = 0 i as required.  Theorem 2.4.6 Suppose that ζ ∈ (t′β )∗ is β-generic and (ζ, δ) 6= 0. Then Yβ (ζ) = M H̄ · ϕx 1ζ . (17) x∈X β ¯ and irreducible as Moreover, each H̄ · ϕx 1ζ is isomorphic to Ȳβ(x) (x(ζ)) an H̄-module. ¯ ∈ Proof. First we will prove the latter part. It is easy to check that x(ζ) ¯ is irreducible. By Lemma t̄∗β(x) and it is β(x)-generic. Hence Ȳβ(x) (x(ζ)) 2.4.5, there exists a surjective H̄-homomorphism ¯ → H̄ϕx 1ζ Ȳβ(x) (x(ζ)) ¯ such that 1x(ζ) ¯ 7→ ϕx 1ζ . This is bijective since Ȳβ(x) (x(ζ)) is irreducible. It follows from Proposition 2.4.3 and Proposition 2.4.4-(i) that H̄ϕx 1ζ = ⊕w∈W̄ β(x) Cϕwx 1ζ . Now the statement follows from Corollary 2.3.12 and Proposition 2.4.3.  For an H-module V , define its W̄ -invariant part by V W̄ = {v ∈ V | wv = v for all w ∈ W̄ }. Then S[t′ ]W̄ acts on V W̄ . For a character χ : S[t′ ]W̄ → C, we set VχW̄ = {v ∈ V W̄ | p · v = χ(p)v for all p ∈ S[t′ ]}. By Proposition 2.4.4-(ii) and Theorem 2.4.6, we have Corollary 2.4.7 Suppose that ζ ∈ (t′ )∗β is β-generic and (ζ, δ) 6= 0. Then Yβ (ζ)W̄ = L x∈X β Yβ (ζ)W̄ [x(ζ)] Moreover, for each x ∈ X β , we have Yβ (ζ)W̄ [x(ζ)] = CQϕx 1ζ 6= 0, where 1 P Q = N ! w∈W̄ w ∈ C[W̄ ]. DDAHA & CFT 2.5 16 Unique irreducible quotients Lemma 2.5.1 Let β ⊢ N and ζ ∈ (t′ )∗β . Then, Yβ (ζ) = M Yβ (ζ)gen ζ′ ζ ′ ∈{w(ζ)|w∈W β } n o β ′ if it is finite. and dim Yβ (ζ)gen ζ ′ = ♯ w ∈ W | w(ζ) = ζ Proof. Use the fact from Proposition 1.3.4 (i), that for any ξ ∈ t′ and P  w ∈ W β , ξ · w1ζ = w(ζ)(ξ)w1ζ + v≺w dv v for some dv ∈ C. Proposition 2.5.2 Let β ⊢ N and ζ ∈ (t′ )∗β . Suppose that dim Yβ (ζ)ζ = 1. Then the H-module Yβ (ζ) has an unique irreducible quotient Lβ (ζ). Proof. Let M be a proper submodule of Yβ (ζ). Then M admits the (genL Yβ (ζ)gen eralized) weight decomposition, thus M ⊂ ζ ′ since the vector ζ ′ 6=ζ 1ζ is a cyclic vector and if Mζgen 6= {0}, then Mζ 6= {0}. Hence the sum of the all proper submodules is the maximal proper submodule of Yβ (ζ).  Proposition 2.5.3 Let β = (β1 , . . . , βm ) ⊢ N and ζ ∈ (t′ )∗β . Set ka = i=1 βi + 1 (a = 1, . . . , m). Suppose that the following conditions hold: Pa−1 (ζ, δ) ∈ / Q≤0 , (ζ, rδ + αka kb ) ∈ / Z≤0 for any 1 ≤ a < b ≤ m, r ∈ Z≥0 , (ζ, rδ − αka kb ) ∈ / Z≤0 for any 1 ≤ a < b ≤ m, r ∈ Z>0 . (18) Then w(ζ) 6= ζ for any w ∈ W β \ {1}, where 1 denotes the unit element of W . In particular Yβ (ζ)gen = C1ζ . ζ Proof. (Step 1) First we prove that w(ζ) 6= ζ for any w ∈ W̄ β \{1}. Suppose that w(ζ) = ζ for w ∈ W̄ β \ {1}. Let p be the smallest number such that w(p) 6= p, and let ka ≤ p < ka+1 . Then w −1 (p) = kb > p(≥ ka ) for some b since w ∈ W̄ β . But then 0 = (ζ − w(ζ), ǫp ) = (ζ, αp w−1 (p) ) = (ζ, αp ka ) + (ζ, αka ,kb ). This implies (ζ, αka,kb ) = (ζ, αka ,p ) ∈ Z≤0 . This contradicts the condition (18). (Step 2) Let x ∈ W β . By Lemma 2.2.4, we can write x = tη · γη−1 · w = γη−1 · tγη (η) · w (η ∈ P̄ , w ∈ W̄ β ). Suppose that x(ζ) = ζ. By (Step 1), P it is sufficent to prove η = 0. Suppose η 6= 0. Putting ǫ = N i=1 ǫi , we have (ζ, ǫ) = (x(ζ), ǫ) = (tγη (η) w(ζ), ǫ) = (ζ, ǫ) + (ζ, δ)(γη (η), ǫ), and thus (γη (η), ǫ) = 0. This implies r := −(γη (η), ǫ1 ) ∈ Z>0 as γη (η) ∈ P̄− . Since tγη (η) w(ζ) = γη (ζ), we have 0 = (tγη (η) w(ζ) − γη (ζ), ǫ1 ) = (γη (η), ǫ1 )(ζ, δ) + (ζ, αw−1(1),γη−1 (1) ). (19) DDAHA & CFT 17 Note that w −1 (1) = ka for some a. Write γη−1 (1) = p and let kb ≤ p < kb+1 . Then (19) leads 0 = −(ζ, rδ − αka ,p ) and thus (ζ, rδ − αka ,kb ) = (ζ, αkb,p ) ∈ Z≤0 . It is easy to see that (20) never occurs under the condition (18). (20)  As a consequence of the results above, we have Corollary 2.5.4 Let β ⊢ N. Suppose that ζ ∈ (t′ )∗β satisfies the condition (18). Then the H-module Yβ (ζ) has a unique irreducible quotient Lβ (ζ). Affine Lie algebras and classical r-matrices We introduce some notations and basic facts on affine Lie algebras, which will be used in the next section. 3.1 (1) Affine Lie algebra of type Am−1 Let ḡ be the Lie algebra slm (C) and g = ḡ ⊗ C[t, t−1 ] ⊕ Ccg ⊕ Cdg be the b (C) associated with ḡ with the commutation relations affine Lie algebra sl m [X ⊗ f, Y ⊗ g] = [X, Y ] ⊗ f g + (X ⊗ f, Y ⊗ g)gcg, , cg ∈ Z(g) [dg, X ⊗ f ] = X ⊗ t df dt for X, Y ∈ ḡ and f, g ∈ C[t, t−1 ], where the invariant bilinear form ( , )g is defined by (X ⊗ f, Y ⊗ g)g = tr(XY ) Rest=0 f dg dt, dt (X ⊗ f, cg)g = (X ⊗ f, dg)g = 0, (cg, dg)g = 1, (cg, cg)g = (dg, dg)g = 0. (21) A Cartan subalgebra h of g is given by h = h̄ ⊕ Ccg ⊕ Cdg, where h̄ is a Carten subalgebra of ḡ. Its dual space is denoted by h∗ = h̄∗ ⊕ Cc∗g ⊕ Cδg, where c∗g and δg denote the dual of dg and cg respectively: c∗g(x) = (dg, x)g, δg (x) = (cg, x)g for any x ∈ h. We write the root system and the set of positive roots of ḡ by Rḡ and Rḡ+ respectively. Then we have the root space decomposition ḡ = h̄ ⊕(⊕γ∈Rḡ ḡγ ). We choose the set {Eγ ∈ ḡγ | γ ∈ Rḡ} of root vectors such that (Eγ , E−γ ) = 1 for each γ ∈ Rḡ+ , and put n̄± = ⊕γ∈Rḡ + ḡ±γ = ⊕γ∈Rḡ + CE±γ . Let {Ha }m−1 a=1 be an orthonormal basis of h̄. DDAHA & CFT 18 The classical r-matrix of ḡ is defined by X X 1 m−1 r̄ = Ha ⊗ Ha + Eγ ⊗ E−γ ∈ ḡ ⊗ ḡ. 2 a=1 γ∈Rḡ (22) + Choose a set {γ0 , . . . , γm−1 } of simple roots of g, then the Weyl group Wg of g is generated by the simple reflections σa corresponding to γa P (a = 1, . . . , m − 1). We put ρḡ = 12 γ∈Rḡ+ γ, and denote its dual by ρˇḡ ∈ h̄. Put ρg = ρḡ + mc∗g, ρˇg = ρˇḡ + mdg. We have a triangular decomposition g = n+ ⊕ h ⊕ n− with n± = n̄± ⊕(ḡ ⊗ C[t±1 ]t±1 ). and put b± = n± ⊕ h. Define the subalgebra g′ of g, and its subalgebra h′ and b′± by g′ = [g, g] = ḡ ⊗ C[t, t−1 ] ⊕ Ccg, 3.2 h′ = h̄ ⊕ Ccg, b′± = n± ⊕ h′ . (23) The universal classical r-matrix We introduce the universal classical r matrix of g and some of its properties, which are used in the next section. Let {Jk }k=1,...,dim ḡ be an orthonormal basis of ḡ. Definition 3.2.1 The universal classical r-matrix r of g is defined by X dimḡ X 1 (Jk ⊗ tn ) ⊗(Jk ⊗ t−n ). r = r̄ + (cg ⊗ dg + dg ⊗ cg) + 2 n≥1 k=1 (24) For each X ∈ g, we put b(X) = [r, X ⊗ 1 + 1 ⊗ X] (X ∈ g). (25) Although the r-matrix itself contains an infinite sum, b(X) always belongs to g ⊗ g, and thus we have the map b : g → g ⊗ g called the Lie bi-algebra structure of g. We let T be the transposition on g ⊗ g and ̟ be the natural projection from g ⊗ g to U(g): T (X ⊗ Y ) = Y ⊗ X, ̟(X ⊗ Y ) = XY (X, Y ∈ g). (26) The following two lemmas play important roles in the next section and can be shown by direct calculations: Lemma 3.2.2 For X ∈ g, we have T b(X) = −b(X), ̟b(X) = [ρˇg , X]. (27) (28) DDAHA & CFT 19 P For a (possibly infinite) sum x = k Xk ⊗ Yk (Xk , Yk ∈ g), we set x1,2 = P x ⊗ 1, x2,3 = 1 ⊗ x and x1,3 = k Xk ⊗ 1 ⊗ Yk . Lemma 3.2.3 The classical Yang-Baxter equation holds: [r1,2 , r2,3 ] + [r1,3 , r2,3 ] + [r1,2 , r1,3 ] = 0.✷ 3.3 (29) Representations of g Recall the category O of left g-modules: its object set (which will be denoted also by O) consists of left g-modules which are finitely generated over g, n+ -locally finite, and h-diagonalizable with finite dimensional weight spaces. We also define the lowest version O† of O: its objects are finitely generated over g, n− -locally finite, and h-diagonalizable with finite dimensional weight spaces. For a weight λ ∈ h∗ , let M(λ) denote the highest-weight Verma module with highest-weight λ, and M † (λ) the lowest-weight Verma module with lowest-weight −λ. Their irreducible quotients are denoted by L(λ) and L† (λ) respectively. Apparently M(λ), L(λ) ∈ O and M † (λ), L† (λ) ∈ O† . For ℓ ∈ C, a g-module is said to be of level ℓ if the center cg acts as the scalar ℓ. In the following we fix a complex number ℓ. Let O(ℓ) (resp. O† (ℓ)) be the full subcategory of O (resp. O† ) whose object set consists of level ℓ (resp. −ℓ) objects in O (resp. O† ). Put h∗ (ℓ) = {λ ∈ h∗ | λ(cg) = ℓ} and let Pg(ℓ) (resp. Pg+ (ℓ)) be the subset of h∗ (ℓ) consisting of integral (resp. dominant integral) weights. Note that L(λ) and L† (λ) become integrable for λ ∈ Pg+ (ℓ) and that Pg(ℓ) is empty unless ℓ in an integer. We also introduce the evaluation module, which do not belong to O nor O† . Let V✷ = Cm be the vector representation of ḡ with natural basis {ua }m notations ēa ∈ h̄∗ a=1 . It is convenient for later use to introduce Pm (a = 1, . . . , m) denoting the weight of ua . Note that a=1 ēa = 0. On the space C[z, z −1 ] ⊗ V✷ , we define a g-module structure of level 0 from the correspondence X ⊗ f (t) 7→ f (z) ⊗ X, dg 7→ z N  ∂ ⊗ id. ∂z  −1 We identify the space C[P̄ ] ⊗ V✷⊗ N with N i=1 C[zi , zi ] ⊗ V✷ , on which g acts diagonally. For a g (resp. ḡ)-module V and a weight λ ∈ h∗ (resp. h̄∗ ) let Vλ denote the weight space of V of weight λ. For λ ∈ h∗ (ℓ), the image of λ under the projection h∗ → h̄∗ is denoted by λ̄ and called the classical part of λ. DDAHA & CFT 20 Construction of H-modules Throughout this paper we use the notation M/a = M/aM, (30) for a Lie algebra a and an a-module M. For A ∈ O(ℓ) and B ∈ O† (ℓ), we put  . F (A, B) = A ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ B g′ , (31) where g′ acts diagonally on the numerator. We will define an action of the degenerate double affine Hecke algebra H on the space F (A, B) using the expression H = C[W ] ⊗ S[t′ ] (see Proposition 1.3.7). 4.1 KZ connections We first define an action of W = W̄ ⋉ P̄ as follows; define the action of W̄ on A ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ B by τ = idA ⊗ τ1 ⊗ τ2 ⊗ idB , where τ1 denote the action on C[P̄ ] and τ2 (w)(v1 ⊗ · · · ⊗ vN ) = vw(1) ⊗ · · · ⊗ vw(N ) , (32) for w ∈ W̄ , and v1 , . . . , vN ∈ V✷ . Let the elements of C[P̄ ] act on A ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ B by usual multiplication. Apparently this action defines an action of W on F (A, B). To define an action of S[t′ ], we introduce the KZ-connections in terms of the universal classical r-matrix of g. Since they do not preserve F (A, B) or A ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ B, we need to consider localizations of these spaces once. Consider the manifold X := (C∗ )N \ { (z1 , . . . , zN ) ∈ (C∗ )N | zi = zj for some i 6= j }. (33) Let R be the sheaf of holomorphic functions on X and let R(U) is the ring of holomorphic functions on an open submanifold U of X. We regard C[P̄ ] ⊗ V✷⊗ N as a subspace of R(X) ⊗ V✷⊗ N . The diagonal action of g on A ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ B extends naturally to the one on A ⊗ R(X) ⊗ V✷⊗ N ⊗ B. Fix a pair of g-modules A, B such that A ∈ O(ℓ), B ∈ O† (ℓ). For i ∈ {0, . . . , N + 1}, let θi : g → g⊗(N +2) be the embedding to the (i + 1)th component. Then θi induces the map g → End C (A ⊗ C[P̄ ] ⊗ V✷ ⊗ B) and is extended to the map g → End C (A ⊗ R(X) ⊗ V✷⊗ N ⊗ B), which are denoted by the same symbol θi . We set θij = θi ⊗ θj and put r(0 i) = θ0 i (r), r(i N +1) = θi N +1 (r) (1 ≤ i ≤ N), (34) DDAHA & CFT 21 which are well-defined operators on A ⊗ R(X) ⊗ V✷⊗ N ⊗ B. On the other hand, for i, j ∈ {1, . . . , N} with i < j, we have formally θij (r) = θij (r̄) + X (zi /zj )n θij (Ω), (35) n≥1 where Ω = r̄ + T (r̄). These operators converge simultaneously only on the region X0 = { (z1 , . . . , zN ) ∈ X | |z1 | < |z2 | < · · · < |zN | } of X to the elements zi /zj r(ij) = θij (r̄) + θij (Ω). (36) 1 − zi /zj of End C (A ⊗ R(X0 ) ⊗ V✷⊗ N ⊗ B). We extend r(i,j) holomorphically to an element in End C (A ⊗ R(X) ⊗ V✷⊗ N ⊗ B). Then Lemma 3.2.3 implies the following: Lemma 4.1.1 The family of operators {r(ij) }0≤i<j≤N +1 satisfies the classical Yang Baxter equations: [r(ij) , r(jk) ] + [r(ik) , r(jk) ] + [r(ij) , r(ik) ] = 0 (0 ≤ i < j < k ≤ N + 1). (37) Definition 4.1.2 We define the KZ-connection ∇i (i = 1, . . . , N) as an element of End (A ⊗ R(X) ⊗ V✷⊗ N ⊗ B) given by ∇i = X r(ji) − 0≤j<i X r(ij) + θi (ρˇg) (38) i<j≤N +1 The following proposition follows from direct calculations using Lemma 4.1.1. Proposition 4.1.3 The followings hold in End (A ⊗ R(X) ⊗ V✷⊗ N ⊗ B) : [∇i , ∇j ] = 0 (1 ≤ i, j ≤ N), τ (w) ∇i τ (w −1 ) = ∇w(i) (w ∈ W̄ ), [∇i , f ] = (ℓ + m)∂i f (f ∈ R(X)). Proposition 4.1.4 Put θ = PN +1 i=0 (39) (40) (41) θi . Then we have [∇i , θ(X)] = (θ ⊗ θi )(b(X)) (42) for each i = 1, . . . , N, and X ∈ g, where θi ⊗ θi (b(X)) means θi ̟(b(X)). In particular, the KZ connections ∇i preserve the subspace g′ (A ⊗ R(X) ⊗ V✷⊗ N ⊗ B) :   ∇i g′ (A ⊗ R(X) ⊗ V✷⊗ N ⊗ B) ⊂ g′ (A ⊗ R(X) ⊗ V✷⊗ N ⊗ B). (43) DDAHA & CFT 22 Proof. For X ∈ g, it is obvious that [r(ij) , (θi + θj )(X)] = θij (b(X)), and thus we have P P P +1 [∇i , N θ (X)] = 0≤j<i θji (b(X)) − i<j≤N +1 θij (b(X)) + θi [ρˇg, X] P P k=0 k = 0≤j≤N +1,j6=i θji (b(X)) + θi ̟(b(X)) = 0≤j≤N +1 θji (b(X)). The second equality follows from (27) and (28). 4.2  The Cherednik-Dunkl operator We generalize the Cherednik-Dunkl operators by combining with the KZ connections to the operators on A ⊗ R(X) ⊗ V✷⊗ N ⊗ B, which turn out to act on F (A, B). For each κ ∈ C and ξ ∈ t̄, the Cherednik-Dunkl operator Dξ = Dξ (κ) ∈ End C (R(X)) is given by Dξ = κ ∂ξ + X α(ξ) α∈R̄+ where ρ̄ = 1 2 P α∈R̄+ 1 − τ1 (sα ) − ρ̄(ξ), 1 − e−α (44) α ∈ t̄∗ . Note that Dξ preserves C[P̄ ]. Proposition 4.2.1 (Cherednik) The correspondence c ξ w f 7→ κ, 7→ Dξ 7→ τ1 (w) 7→ f × (ξ ∈ t̄), (w ∈ W̄ ), (f ∈ R(X)) (45) defines an action of H on R(X) and on C[P̄ ]. Let us generalize the Cherednik-Dunkl operators to our case. For ξ ∈ t̄, we set ∇ξ = N X ǫi (ξ)∇i (46) i=1 and define the operator Dξ = Dξ (κ) ∈ End (A ⊗ R(X) ⊗ V✷⊗ N ⊗ B) by Dξ := ∇ξ + X α∈R̄+ α(ξ) 1 − τ (sα ) m−1 − ρ̄(ξ) − ∂ξ logG, −α 1−e 2m where G = Πα∈R̄ (1 − e−α ) ∈ C[P̄ ]. (47) DDAHA & CFT 23 Theorem 4.2.2 (i) The correspondence c ξ w f 7 ℓ + m, → 7→ Dξ (ξ ∈ t̄), 7→ τ (w) (w ∈ W̄ ), 7→ f × (f ∈ R(X)) (48) defines an action of H on A ⊗ R(X) ⊗ V✷⊗ N ⊗ B. (ii) The subspaces A ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ B and g′ (A ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ B) are preserved by the above action. Therefore the correspondence (48) induces an action of H on the space F (A, B). Proof. (i) By Proposition 4.1.3, the relations between the operators ∇ξ , w and f are same with the ones between (ℓ + m)∂ξ , w and f . Therefore the statement is shown exactly as Proposition 4.2.1. (ii) The fact that Dξ preserves A ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ B follows from direct calculations. The rest is straightforward from the definition of the action and Proposition 4.1.4.  In the rest of this paper, we put κ = ℓ + m. It is easily checked that our construction is functorial in the following sense: for g-homomorphisms f1 : A → A′ and f2 : B → B ′ between g-modules, the induced maps f1∗ : F (A, B) → F (A′ , B), f2∗ : F (A, B) → F (A, B ′ ) (49) are both H-homomorphisms. Therefore we have the bifunctor from O(ℓ)× O† (ℓ) to the category of H(κ)-modules. Structure of H-modules Our next aim is to study H-module structures of F (A, B) for A ∈ O(ℓ) and B ∈ O† (ℓ). For each µ ∈ h∗ (ℓ), we can define the right exact functor Fµ (·) = F (M(µ), · ) (50) from O† (ℓ) to the category of H-modules. We put M(λ, µ) = Fµ (M † (λ)), V(λ, µ) = Fµ (L† (λ)), for each λ, µ ∈ h∗ (ℓ). When λ, µ ∈ Pg+ (ℓ), the space V(λ, µ) is related to the dual space of conformal blocks (see Proposition 5.1.3 and [7]). DDAHA & CFT 5.1 24 Fundamental properties In this subsection, we study some properties of the bifunctor F , which will be used later. Let v(µ) and v † (λ) denote the highest (and lowest) weight vector of M(µ) and M † (λ) respectively. Lemma 5.1.1 For A ∈ O(ℓ), B ∈ O† (ℓ), λ ∈ h∗ (ℓ), and µ ∈ h∗ (ℓ), the natural inclusions Cv(µ) → M(µ) and Cv † (λ) → M † (λ) induce isomorphisms F (M(µ), B) F (A, M † (λ)) ∼ = Cv(µ) ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ B/b′+ , ∼ = A ⊗ C[P̄ ] ⊗ V✷⊗ N ⊗ Cv † (λ)/b′− . (51) Proof. Follows from the following well-known g-isomorphism for any gmodule V : M(µ) ⊗ V ∼ (52) = U(g) ⊗ (Cv(µ) ⊗ V ) U (b+ ) and its lowest version, where on the left-hand-side of (52), g acts diagonally and on the right-hand-side by left multiplication.  Again by (52), there is an isomorphism of vector spaces M(λ, µ) ∼ = Cv(µ) ⊗(C[P̄ ] ⊗ V✷⊗ N )λ−µ ⊗ Cv † (λ). Notice that the space on the right-hand-side is an H-submodule of M(µ) ⊗(C[P̄ ] ⊗ V✷⊗ N ) ⊗ M † (λ) with respect to the action (48). Hence we have Proposition 5.1.2 For λ, µ ∈ h∗ (ℓ), we have M(λ, µ) ∼ = Cv(µ) ⊗(C[P̄ ] ⊗ V✷⊗ N )λ−µ ⊗ Cv † (λ) (53) as H-modules. Proposition 5.1.3 For λ, µ ∈ Pg+ (ℓ), the natural projection M(µ) → L(µ) induces an H-isomorphism Fµ (L† (λ)) ∼ = F (L(µ), L† (λ)). (54) Proof. It is obvious that the induced map F (M(µ), L† (λ)) → F (L(µ), L† (λ)) is surjective. To show the injectivity, recall that the maximal submodule of M(µ) is generated by the singular vectors va ∈ M(µ)σa ◦µ (a = 0, . . . , m − 1) satisfying Xva = 0 for any X ∈ n+ , where σa ◦ µ = σa (µ + ρg) − ρg. By elementary calculations, it can be checked DDAHA & CFT 25 that, for any number n ≥ 0, there exists a non-zero constant cn such n that Eγna E−γ v = cn va , where Eγ denotes the root vector corresponding a a to γ ∈ Rg . Put V = C[P ] ⊗ V✷⊗ N ⊗ L† (λ). It is enough to show that U(g)va ⊗ V ⊂ g(M(µ) ⊗ V ) for each a = 0, . . . , m, which is reduced to showing that va ⊗ u ∈ g(M(µ) ⊗ V ) (55) for any u ∈ V . Since V is integrable and thus n+ -locally finite, there exists a number n > 0 such that Eani u = 0, and we have n n cn va ⊗ u = va ⊗ Eγna E−γ u ≡ E−γ Eγna vi ⊗ u ≡ 0 mod g(M(µ) ⊗ V ), a a that implies (55). 5.2  Isomorphisms to induced modules We shall describe the H-module structure of M(λ, µ), which will be shown to be isomorphic to an induced module Yβ (ζ) = H(κ) ⊗H̄β C1ζ (see §2.1) associated to an appropriate parameter (β, ζ) determined by λ, µ ∈ h∗ (ℓ). Recall that ēa ∈ h̄∗ (a = 1, . . . , m) denotes the weight of the standard basis ua of V✷ . Let us associate an ordered partition β of N and an element ζ ∈ (t′ )∗ to each pair λ, µ ∈ h∗ . Assume that λ, µ ∈ h∗ (ℓ) satisfy λ̄ − µ̄ ∈ P wt (V✷⊗ N ). Then λ̄ − µ̄ = m a=1 βa ēa for some βa ∈ Z≥0 (a = 1, . . . , m) Pm satisfying a=1 βa = N. Thus we define an ordered partition βλ,µ of N by βλ,µ = (β1 , . . . , βm ). (56) Here, we allowed appearance of zeros as elements of an ordered partitions. We keep the assumption that λ̄ − µ̄ ∈ wt(V✷⊗ N ). Put µa = (µ, ēa ) (a = 1, . . . , m) and consider the set Yλ,µ = Yλ̄,µ̄ := { (a, p) ∈ Z×C | a = 1, . . . , m; p = µa +1, µa +2, . . . , µa +βa }, (57) which we represent just by Y below. A bijection T : Y → {1, . . . , N} is called a tableau on Y . We let T (λ̄, µ̄) denote the set of tableaux on Y . Define T0 ∈ T (λ, µ) by T0 (a, p) = a−1 X βk + p − µa . (58) k=1 Now, define ζ̄λ̄,µ̄ ∈ t̄∗ and ζλ,µ ∈ (t′ )∗ by ζ̄λ̄,µ̄ N X 1 2 = (p − a)ǫT0 (a,p) + (m − N) ǫk 2m k=1 (a,p)∈Y X ζλ,µ = ζ̄λ̄,µ̄ + κc∗ (59) (60) DDAHA & CFT 26 For T ∈ T (λ̄, µ̄), let wT denote the inverse image of T ∈ T (λ̄, µ̄) under the isomorphism W̄ → T (λ̄, µ̄) w 7→ wT0 : (a, p) 7→ w(T0 (a, p)). Note that N X 1 2 wT (ζ̄λ̄,µ̄ ) = (p − a)ǫT (a,p) + (m − N) ǫk . 2m k=1 (a,p)∈Y X (61) The following lemma can be checked easily. Lemma 5.2.1 If λ̄ − µ̄ ∈ wt(V✷⊗ N ), then ζλ,µ (resp. ζ̄λ̄,µ̄ ) belongs to (t′ )∗βλ,µ (resp. t̄∗βλ,µ ) (see (11) for the definition of (t′ )∗β ). Let u(µ) and u† (λ) be the highest and lowest weight vector in M(µ) and M † (λ) respectively, and put z β1 }| { β2 z }| z { βm }| { uλ,µ = v(µ) ⊗ u1 ⊗ · · · ⊗ u1 ⊗ u2 ⊗ · · · ⊗ u2 ⊗ · · · ⊗ um ⊗ · · · ⊗ um ⊗ v † (λ). (62) By direct calculations, we have the following lemma. Lemma 5.2.2 For λ, µ ∈ h∗ (ℓ) such that λ̄ − µ̄ ∈ wt(V✷⊗ N ), the vector uλ,µ is a weight vector with the weight ζλ,µ : Dξ uλ,µ = ζλ,µ (ξ)uλ,µ for ξ ∈ t. (63) By Proposition 5.1.2 and Lemma 5.2.2, we get the following conclusion: Proposition 5.2.3 For any λ, µ ∈ h∗ (ℓ), we have a unique H-module isomorphism M(λ, µ) ∼ = ( Yβλ,µ (ζλ,µ ) 0 if λ̄ − µ̄ ∈ wt(V✷⊗ N ), otherwise, (64) which sends uλ,µ to 1ζλ,µ . 5.3 Finite analogue of the functor F It is possible to construct the “finite version” of our functor F , which is easier to study and has some suggestive properties. Consider the category Ō (resp. Ō∗ ) of highest (resp. lowest) weight ḡ-modules. For ḡ-modules A ∈ Ō and B ∈ Ō∗ , we put  F̄ (A, B) := A ⊗ V✷⊗ N ⊗ B . ḡ. (65) DDAHA & CFT 27 and consider the operator D̄i :=  X  0≤j<i   X 1 1 − + θi (ρˇḡ). θij (r̄) + θji (r̄) + 2m 2m i<j≤N +1 (66) It can be shown that D̄i acts on F̄(A, B) and the correspondence ǫiˇ 7→ D̄i w 7→ τ2 (w) i = 1, . . . , N, w ∈ W̄ (67) defines an action of H̄ on F̄ (A, B). For each µ ∈ h̄∗ , we define a functor from the category Ō∗ to the category of finite dimensional H̄-modules by F̄µ̄ (·) = F̄(M̄ (µ̄), ·), (68) and put F̄µ̄ (M̄ † (λ̄)) = M̄(λ̄, µ̄), F̄µ̄ (L̄† (λ̄)) = V̄(λ̄, µ̄), where M̄(λ̄) denotes the highest weight left Verma module of ḡ with highest weight λ̄ ∈ h̄∗ etc. As an analogue of Proposition 5.1.3 we have V̄(λ̄, µ̄) ∼ = F̄ (L̄(µ̄), L̄† (λ̄)) if λ̄, µ̄ ∈ h̄∗ are both dominant integral. An analogue of Proposition 5.2.3 also holds: Proposition 5.3.1 For any λ̄, µ̄ ∈ h̄∗ , we have the following H̄-module isomorphism: M̄(λ̄, µ̄) ∼ = ( Ȳβλ̄,µ̄ (ζ̄λ̄,µ̄ ) 0 if λ̄ − µ̄ ∈ wt(V✷⊗ N ), otherwise, (69) where βλ̄,µ̄ and ζ̄λ̄,µ̄ are defined by the similar formulas as (56) and (59) respectively. Under finite situations, standard arguments using the infinitesimal characters deduce the following: Proposition 5.3.2 Suppose that µ̄ ∈ h̄∗ satisfies / {−1, −2, . . .} (µ̄ + ρḡ, γ) ∈ for all γ ∈ Rḡ+ , then the functor F̄µ̄ is exact. Now, let us suppose that λ̄, µ̄ are both dominant integral. Then L̄(λ̄) is finite dimensional and the dimension of F̄µ̄ (L̄(λ̄)) is calculated using Shur-Weyl reciprocity law and turns out to equal the number of standard tableaux on the skew Young diagram Yλ̄,µ̄ : A tableau T ∈ T (λ̄, µ̄) on Yλ̄,µ̄ is called a standard tableau if T satisfies the following two conditions: T (a, p) < T (a, p + 1) if (a, p), (a, p + 1) ∈ Yλ̄,µ̄ , T (a, p) < T (a + 1, p) if (a, p), (a + 1, p) ∈ Yλ̄,µ̄ . DDAHA & CFT 28 Let Ts (λ̄, µ̄) denote the set of standard tableaux on Yλ̄,µ̄ (then dim V(λ̄, µ̄) = #Ts (λ̄, µ̄)), Note that wT ∈ W̄ βλ,µ for T ∈ Ts (λ, µ). The following proposition can be proved by using Proposition 1.4.3 and Proposition 2.5.3. Proposition 5.3.3 Let λ̄, µ̄ ∈ h̄∗ be both dominant integral. Then V̄(λ̄, µ̄) is the unique irreducible quotient of M̄(λ̄, µ̄). Moreover, its basis is given by {ϕwT 1ζλ̄,µ̄ }T ∈Ts(λ̄,µ̄) . In particular S[ t̄ ] acts semisimply on V̄(λ̄, µ̄). 5.4 Dominant integral case As a consequence of Proposition 5.2.3, we have that the H-module M(λ, µ) is irreducible if ζλ,µ is βλ,µ -generic (Definition 2.4.1). Let us consider the case where λ and µ are both dominant integral. The following statement is a corollary of Corollary 2.5.4: Corollary 5.4.1 If µ ∈ Pg+ (ℓ), then M(λ, µ) has a unique irreducible quotient. Proof. It can be checked that the condition µ ∈ Pg+ (ℓ) implies the condition (18) in Corollary 2.5.4 for ζλ,µ .  Conjecture 5.4.2 Let λ, µ ∈ h∗ (ℓ) be both dominant integral. Then V(λ, µ) is the unique irreducible quotient of M(λ, µ). Moreover S[ t′ ] acts semisimply on V(λ, µ). Symmetric part In the rest of this paper we focus on the W̄ -invariant part of V(λ, µ) for λ, µ ∈ Pg+ (ℓ) and attempt to give a description corresponding to Corollary 2.4.7. 6.1 Specialized characters As an approach to investigate the H-modules, we consider their characters with respect to the operator ( ) N X 1 1 L1 := Di − N(m2 − N) ∈ S[ t̄ ]W̄ . ℓ + m i=1 2m (70) To this end, we introduce “the polynomial part” of H and Fµ (B) (see §1.3). Put Fµ≥0 (B) = Cv(µ) ⊗ C[P̄ ≥0 ] ⊗ V✷⊗ N ⊗ B/b′+ . (71) DDAHA & CFT 29 Then Fµ≥0 (B) has the H≥0 -module structure. For a vector space M with a semisimple action of L1 with a finite dimensional eigen space decomposition, we put ch M = TraceM q L1 (72) and call ch M the specialized character of M, where q is a parameter. It is easily checked that ch Fµ≥0 (B) is well-defined for any B ∈ O† (ℓ). Put M≥0 (λ, µ) = Fµ≥0 (M † (λ)) and V ≥0 (λ, µ) = Fµ≥0 (L† (λ)). The same argument as Lemma 5.1.2 shows that M≥0 (λ, µ) ∼ = Cv(µ) ⊗(C[P̄ ≥0 ] ⊗ V✷⊗ N )λ−µ ⊗ Cv † (λ), as an H-module. On the right-hand-side, we have L1 ≡ △λ̄ − △µ̄ + N X ∂i , (73) i=1 1 2(ℓ+m) ≥0 where △λ̄ =   (λ̄, λ̄) + 2(ρ̄g, λ̄) . ¿From (73), the specialized char- acters of M (λ, µ) and M≥0 (λ, µ)W̄ are calculated as ! N q △λ̄ −△µ̄ ch M (λ, µ) = , (q)N β " λ,µ # N q △λ̄ −△µ̄ , ch M≥0 (λ, µ)W̄ = (q)N βλ,µ ≥0 respectively, where (q)N = (1 − q)(1 − q 2 ) · · · (1 − q N ) and N βλ,µ ! N! = , β1 !β2 ! · · · βm ! " N βλ,µ # = (q)N . (q)β1 (q)β2 · · · (q)βm It is natural to attempt to construct a resolution of V(λ, µ) by induced modules to calculate the character for V(λ, µ). Recall the BGG resolution for L† (λ), which is the exact sequence 0 ← L† (λ) ← M † (λ) ← C1 ← · · · ← Ci ← · · · (74) of g-modules with Ci = ⊕ w∈Wg (i) M † (w ◦ λ), where Wg(i) is the subset of the Weyl group of g consisting of elements of length i. Sending by the functor Fµ (resp. Fµ≥0 ), we have the complex of H (resp. H≥0 )-modules respectively: 0 ← V(λ, µ) ← M(λ, µ) ← C1 ← · · · ← Ci ← · · · , 0 ← V ≥0 (λ, µ) ← M≥0 (λ, µ) ← C1≥0 ← · · · ← Ci≥0 ← · · · , (75) (76) DDAHA & CFT 30 where we put M≥0 (λ, µ) = Fµ≥0 (M † (λ)) and V ≥0 (λ, µ) = Fµ≥0(L† (λ)). Note that each Ci or Ci≥0 is a direct sum of induced modules: Ci = ⊕ w∈Wg (i) M(w ◦ λ, µ), Ci≥0 = ⊕ w∈Wg (i) M≥0 (w ◦ λ, µ). (77) Therefore if the sequence (76) is exact, we can calculate the character of V ≥0 (λ, µ). It can be shown the above sequences are exact when κ is large enough, but in general case it is still conjecture: Conjecture 6.1.1 The sequences (75) and (76) are exact. Note that the above conjecture leads to the formula chV ≥0 (λ, µ) W̄ = X (−1) l(w) (w◦λ)(dg ) q w∈Wg 6.2 " N βw◦λ,µ # q △λ̄ −△µ̄ , (q)N (78) Main conjecture In the following we fix a pair (λ, µ) ∈ Pg+ (ℓ) × Pg+ (ℓ) satisfying λ̄ − µ̄ ∈ wt (V✷⊗ N ). Recall that we associated a partition βλ,µ of N such that P λ̄ − µ̄ = m a=1 βa ēa , and a diagram Yλ,µ = {(a, p) ∈ Z × C | a = 1, . . . , m ; p = µa + 1, µa + 2 . . . , µa + βa } . (79) where µ̄a = (µa , ēi ). As in §5.3, let T (λ, µ) (resp. Ts (λ, µ)) denote the set of tableaux (resp standard tableaux) on Y = Yλ,µ . (n) (n) (n) For T ∈ T (λ, µ) and a number n ≤ N, let βT = (βT 1 , . . . , βT m ) (n) denote the partition of n corresponding to the diagram YT = {(a, p) ∈ (n) (n) Yλ,µ | T (a, p) ≤ n}: namely, βT a = #{p ∈ Z + µa | (a, p) ∈ YT }. Put (n) λT = m  X (n) µa + βT a − a=1 (n)  n ēa + ℓc∗g. m Then it is checked that λT belongs to Pg(ℓ). A standard tableau T ∈ (1) (2) (N ) Ts (λ, µ) is said to be ℓ-restricted if all the weights λT , λT , · · · , λT belong to Pg+ (ℓ). We let Ts(ℓ) (λ, µ) denote the set of ℓ-restricted standard tableaux on Y . For T ∈ T (λ, µ), let wT ∈ W̄ β be the corresponding element. Direct calculations imply the following lemma: Lemma 6.2.1 Let T ∈ Ts(ℓ) (λ, µ). DDAHA & CFT 31 (i) For i ∈ {1, . . . , N − 1}, let T (a, p) = i and T (a′ , p′ ) = i + 1. Then we have −(κ − 1) ≤ (wT (ζλ,µ ), αi ) ≤ −2 if a > a′ , (wT (ζλ,µ ), αi ) = −1 if a = a′ , (80) 1 ≤ (wT (ζλ,µ ), αi ) ≤ κ − 2 if a < a′ . (ii) For any α ∈ R̄+ , we have (wT ζλ,µ , α) ≤ κ − 2. (81) Recall that we associated ηw ∈ P̄− (w) for each w ∈ W̄ β in Definition 2.3.2. Lemma 6.2.2 For each T ∈ Ts(ℓ) (λ, µ), we have ηwT = N −1 X di (T )(ǫˇi+1 + · · · + ǫˇN ). (82) i=1 Here di (T ) = ( if a < a′ , if a ≥ a′ 1 0 with T (a, p) = i and T (a′ , p′ ) = i + 1. Remark 6.2.3 The function di : Ts(ℓ) (λ, µ) → N can be identified with the so-called H-function in the RSOS model (see [8]). Let d(T ) = ηwT (ǫ1 + · · · + ǫN ) and define Fλ,µ (q) = X q d(T ) . (83) (ℓ) T ∈Ts (λ,µ) By the above remark, the following holds by using standard arguments in the solvable lattice model (see [8],[9]): Theorem 6.2.4 Fλ,µ (q) = X (−1) l(w) (w◦λ)(dg ) q w∈Wg " N βw◦λ,µ # . (84) Combining with Conjecture 6.1.1, we have a simple formula ch V ≥0 (λ, µ)W̄ = Fλ,µ (q) · β q △λ̄ −△µ̄ . (q)N Set Xℓ λ,µ = {tη wT | T ∈ Ts(ℓ) (λ, µ), η ∈ P̄− (wT )} ⊆ X βλ,µ . (85) DDAHA & CFT 32 P Theorem 6.2.5 Let λ, µ ∈ Pg+ (ℓ) and Q = N1 ! w∈W̄ w ∈ C[W̄ ]. Then β for each x ∈ Xℓ λ,µ , the element Qϕx · 1ζλ,µ is non-zero. Moreover the set {Qϕx · 1ζλ,µ } βλ,µ is linearly independent: x∈Xℓ V(µ, λ)W̄ ⊇ M CQϕx · 1ζλ,µ . βλ,µ x∈Xℓ In particular, we have chV ≥0 (µ, λ)W̄ ≥ Fλ,µ (q) · q △λ̄ −△µ̄ . (q)N β Proof. Take any x ∈ Xℓ λ,µ . It is easy to see x(ζλ,µ ) ∈ t̄∗β(x) . Direct calculations using Lemma 6.2.1 and Lemma 6.2.2 implies that x(ζλ,µ ) ∈ t̄∗β(x) is β(x)-generic, (86) and (ζλ,µ , α) 6= 0, ±1 for any α ∈ S(x). In particular, ϕx is invertible on 1ζλ,µ . (87) By Lemma 2.4.5, we have si ϕx 1ζλ,µ = ϕx 1ζλ,µ for all αi ∈ Π̄β(x) . Thus we have an H-homomorphism Ȳβ(x) (x(ζλ,µ )) → H̄ϕx 1ζλ,µ (88) defined via 1x(ζλ,µ ) 7→ ϕx · 1ζλ,µ . It follows from (86) that Ȳβ (x)(x(ζλ,µ )) is irreducible, and thus Ȳβ (x)(x(ζλ,µ )) ∼ = H̄ϕx 1ζλ,µ . Therefore Qϕx 1ζλ,µ 6= 0. Next let us prove that {Qϕx 1ζλ,µ } βλ,µ is linearly independent. x∈Xℓ Note that µ ∈ Pg+ (ℓ) implies that βλ,µ and ζλ,µ satisfies the condition (18) in Proposition 2.5.3. Hence we have Yβλ,µ (ζλ,µ )gen ζλ,µ = C1ζλ,µ . (89) Combining with (87), it follows that β the map Xℓ λ,µ → (t′ )∗β given by x 7→ x(ζλ,µ ) is injective. β For any x ∈ Xℓ λ,µ , it can be proved from Lemma 6.2.1 and Lemma 6.2.2 that (x(ζλ,µ ), αi ) ≤ 0 for any i = 1, . . . , N − 1. DDAHA & CFT 33 Therefore β the map Xℓ λ,µ → (t′ )∗β /W̄ given by x 7→ [x(ζλ,µ )] is injective. Hence {Qϕx · 1ζλ,µ } β x∈Xℓ λ,µ is linearly independent.  The specialized character formula (85) and the above Theorem 6.2.5 imply the following: Conjecture 6.2.6 For λ, µ ∈ Pg+ (ℓ), we have V(µ, λ)W̄ = M (ℓ) T ∈Ts (λ,µ) M CQϕtη ·wT · 1ζλ,µ . (90) η∈P̄− (wT ) References [1] I. Cherednik, Elliptic Quantum Many-Body Problem and Double Affine Kniznik-Zamolodohikov Equation, Commun. Math. Phys., 169, 441-461 (1995) [2] A. Matsuo, Integrable connections related to zonal spherical functions. Invent. Math. 110 95-121(1992). [3] E. M. Opdam, Harmonic analysis for certain representation of graded Hecke algebras, Acta. Math. , 175, 75-121 (1995) [4] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc., 2, 599-685 (1989) [5] J. D. Rogawski, On modules over the Hecke Algebra of a p-adic group, Invent. math., 79, 443-465 (1985) [6] I. Cherednik, Special Bases of Irreducible Representations of a Degenerate Affine Hecke Algebra, Func. Anal. Appl, 20:1, 87-89 (1986) [7] A. Tsuchiya, K. Ueno and H. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Stud. Pure Math., 19, 459-566 (1989) [8] M. Jimbo, T. Miwa and M. Okado, Local State Probabilities Of Solvable Lattice Models, Nucl. Phys. B300[FS22], 74-108 (1988) [9] G. E. Andrews, R. J. Baxter, P. J. Forrester, Eight-vertex SOS model and generalized Rogers-Ramanujan type identities, J. Stat. Phys. 35, 193-266 (1984) DDAHA & CFT 34 [10] T. Arakawa, T. Nakanishi, K. Ohshima, A. Tsuchiya, Spectral decomposition of path space in solvable lattice model, Commun. Math. Phys. 181, 157-182 (1996) (Tomoyuki Arakawa) Graduate School of Mathematics, Nagoya University, Japan e-mail: tarakawa@math.nagoya-u.ac.jp (Takeshi Suzuki) Research Institute for Mathematical Sciences, Kyoto University, Japan e-mail: takeshi@kurims.kyoto-u.ac.jp (Akihiro Tsuchiya) Graduate School of Mathematics, Nagoya University, Japan e-mail: tsuchiya@math.nagoya-u.ac.jp