We study a class of representations called ``calibrated representations'' of
the degenerate double affine Hecke algebra and those of the rational Cherednik
algebra of type ${\mathrm{GL}}_n$. We give a realization of calibrated
irreducible modules as spaces of coinvariants constructed from integrable
modules over the affine Lie algebra $\hat{\mathfrak{gl}}_m$. Moreover, we give
a character formula of these irreducible modules in terms of a level-restricted
Kostka polynomials. These results were conjectured by Arakawa, Suzuki and
Tsuchiya based on the conformal field theory. The proofs using recent results
on the representation theory of the double affine Hecke algebras will be
presented in the forthcoming papers.Comment: 8 page

Suzuki, Takeshi

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 343 (2006) 383–386http://france.elsevier.com/direct/CRASS1/AlgebraDouble affine Hecke algebras, conformal coinvariantsand Kostka polynomialsTakeshi SuzukiResearch Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, JapanReceived 6 May 2005; accepted after revision 23 August 2006Available online 22 September 2006Presented by Michel DufloAbstractWe study a class of representations called ‘calibrated representations’ of the rational and trigonometric double affine Heckealgebras of type GLn. We give a realization of calibrated irreducible modules as spaces of coinvariants constructed from integrablemodules over the affine Lie algebra ĝlm. We also give a character formula of these irreducible modules in terms of a generalizationof Kostka polynomials. These results are conjectured by Arakawa, Suzuki and Tsuchiya based on the conformal field theory. Theproofs using recent results on the representation theory of the double affine Hecke algebras will be presented in the forthcomingpapers. To cite this article: T. Suzuki, C. R. Acad. Sci. Paris, Ser. I 343 (2006).© 2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméAlgèbres de Hecke affine double, co-invariants conformes et polynômes de Kostka. Nous étudions la classe de représen-tations, qui s’appelle les représentations calibrées, de l’algèbre de Hecke affine double rationnelle/trigonométrique de type GLn.Nous réalisons les modules simples calibrés comme des espaces de co-invariants construits de modules intégrables sur l’algèbrede Lie affine ĝlm. En plus, nous donnons une formule de caractère de ces modules simples en terme d’une généralisation des poly-nômes de Kostka. Ces résultats sont conjecturés par Arakawa, Suzuki et Tsuchiya en se basant sur la théorie de champs conformes.Pour leur démonstration, nous utilisons les résultats récents de la théorie des représentations de l’algèbre de Hecke affine double.Les détails seront donnés dans des publications ultérieures. Pour citer cet article : T. Suzuki, C. R. Acad. Sci. Paris, Ser. I 343(2006).© 2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. The affine Lie algebra and the double affine Hecke algebrasThroughout this article, we use the notation [i, j ] = {i, i + 1, . . . , j} for i, j ∈ Z. Let F be an algebraically closedfield of characteristic 0.Fix m ∈ Z1. Let g denote the Lie algebra glm(F) consisting of all the m × m matrices over F. Let ĝ =g ⊗ F[t, t−1] ⊕ Fc be the associated affine Lie algebra with the central element c and the commutation relation[a⊗ t i , b⊗ tj ] = [a, b]⊗ t i+j +Tr(ab)iδi+j,0c for a, b ∈ g, i, j ∈ Z. Put g[t] = g⊗F[t] and g[t, t−1] = g⊗F[t, t−1],E-mail address: takeshi@kurims.kyoto-u.ac.jp (T. Suzuki).1631-073X/$ – see front matter © 2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2006.08.009384 T. Suzuki / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 383–386which are Lie subalgebras of ĝ. Let h denote the Cartan subalgebra of g consisting of diagonal matrices. Put ĥ =h ⊕ Fc, which gives a Cartan subalgebra of ĝ. Denote the dual space of h (resp. ĥ) by h∗ (resp. ĥ∗ = h∗ ⊕ Fc∗, wherec∗ is the dual vector of c). A ĝ-module M is said to be of level  ∈ F if the center c acts as a scalar  on M . For κ ∈ Fand λ ∈ h∗, let Lκ(λ) (resp. L†κ (λ)) denote the highest weight (resp. lowest weight) irreducible ĝ-module with highestweight λ+ (κ −m)c∗ (resp. lowest weight −λ− (κ −m)c∗). We naturally identify h∗ with Fm, and introduce its sub-spaces Xm = Zm, X+m = {(λ1, . . . , λm) ∈ Xm | λ1  λ2  . . .  λm} and X+m(κ) = {(λ1, . . . , λm) ∈ X+m | κ − m − λ1 +λm ∈ Z0}. Note that Lκ(λ) and L†κ(λ) are integrable for λ ∈ X+m(κ), and that X+m(κ) is empty unless κ − m ∈ Z0.Let V = Fm denote the basic representation of g. Put V [x, x−1] = V ⊗ F[x, x−1] (resp. V [x] = V ⊗ F[x]), andregard V [x, x−1] (resp. V [x]) as a g[t, t−1]-module (resp. g[t]-module) through the correspondence a ⊗ t i → a ⊗ xi(a ∈ g, i ∈ Z).Fix n ∈ Z2. In the rest, we write F[ z ] (resp. F[ z±1]) for the polynomial ring F[z1, . . . , zn] (resp. the Laurent poly-nomial ring F[z±11 , . . . , z±1n ]) of n-variables. Let E denote the n-dimensional vector space with the basis {ε∨i }i∈[1,n].Introduce the bilinear form (|) on E by (ε∨i |ε∨j ) = δij . Let E∗ =⊕ni=1 Fεi be the dual space of E, where εi isthe dual vector of ε∨i . The natural pairing is denoted by 〈|〉 :E∗ × E → F. Put αij = εi − εj , α∨ij = ε∨i − ε∨j andαi = εi − εi+1, α∨i = ε∨i − ε∨i+1. Then R = {αij | i, j ∈ [1, n], i = j} and R+ = {αij ∈ R | i < j} give a set of rootsand a set of positive roots of type An−1 respectively. Let W denote the Weyl group associated with the root system R,which is isomorphic to the symmetric group of degree n. Denote by sα the reflection in W corresponding to α ∈ R.We write si = sαi and sij = sαij . Put P =⊕i∈[1,n] Zεi . Let S(E) (resp. S(E∗)) denote the symmetric algebra of E(resp. E∗), which is identified with F[ ε∨] (resp. F[ ε ]).Let κ ∈ F. The rational double affine Hecke algebra (or the rational Cherednik algebra) Hκ of type GLn is theunital associative F-algebra which is generated by the algebras S(E∗), FW and S(E), and subjects to the followingdefining relations:sihsi = si(h) (i ∈ [1, n − 1], h ∈ E), siζ si = si(ζ ) (i ∈ [1, n − 1], ζ ∈ E∗),[h, ζ ] = κ〈ζ | h〉 +∑α∈R+〈α | h〉〈ζ | α∨〉sα (h ∈ E, ζ ∈ E∗).It is known by Cherednik that Hκ ∼= S(E∗) ⊗ FW ⊗ S(E) as a vector space via the natural multiplication.For λ ∈ Zm = Xm, we write λ |= n when ∑mi=1 λi = n and λi ∈ Z0 for all i ∈ [1,m]. Let λ ∈ X+m such that λ |= n.We sometimes identify λ with the Young diagram {(a, b) ∈ Z2 | a ∈ [1,m], b ∈ [1, λa]} ⊂ Z2. Let Sλ denote theirreducible module of W associated with the partition λ. Set Δκ(λ) = Hκ ⊗FWS(E) Sλ, where we let S(E) act on Sλthrough the augmentation map given by ε∨i → 0 (i ∈ [1, n]). It is known that the Hκ -module Δκ(λ) has a uniquesimple quotient [4], which we will denote by Lκ(λ).The trigonometric double affine Hecke algebra (or the degenerate double affine Hecke algebra) H̃κ of type GLn isthe unital associative F-algebra which is generated by the algebras FP , FW and S(E), and subjects to the followingdefining relations:sih = si(h)si − 〈αi | h〉 (i ∈ [1, n − 1], h ∈ E), sieηsi = esi (η) (i ∈ [1, n − 1], η ∈ P),[h, eη] = κ〈η | h〉eη + ∑α∈R+〈α | h〉eη − esα(η)1 − e−α sα (h ∈ E, η ∈ P),where eη denotes the element of FP corresponding to η ∈ P .It is known by Cherednik that H̃κ ∼= FP ⊗ FW ⊗ S(E). The subalgebra FW · S(E) ⊂ H̃κ is called the degenerateaffine Hecke algebra Haff. Let λ,μ ∈ X+m such that λ − μ |= n. Let Sλ/μ denote the irreducible module of Haffassociated with the skew diagram λ/μ [3,5]. Set Δ̃κ(λ,μ) = H̃κ ⊗Haff Sλ/μ. When λ,μ ∈ X+m(κ), it is known that theH̃κ -module Δ̃κ(λ,μ) has a unique simple quotient [1], which we will denote by L̃κ(λ,μ).2. The trigonometric double affine Hecke algebra and conformal field theoryFor a Lie algebra a and an a-module M , we write Ma = M/aM . Let N be a highest weight ĝ-module of levelκ − m and let M be a lowest weight ĝ-module of level −κ + m. DefineC̃(N,M) = (N ⊗ V [x1, x−11 ] ⊗ . . . ⊗ V [xn, x−1n ] ⊗ M)g[t,t−1].T. Suzuki / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 383–386 385Remark 2.1. The ring F[x±1] = F[x±11 , . . . , x±1n ] can be identified with the coordinate ring of the affine variety T =(F\ {0})n, and acts on C̃(N,M). Let A denote the localization of F[x±1] at the diagonal set Δ = ⋃i<j {(x1, . . . , xn) ∈T | xi = xj }. Then, it follows that the A-module A ⊗F[x±1] C̃(N,M) admits an integrable connection (called theKnizhnik–Zamolodchikov connection), and gives a vector bundle on the affine variety T \ Δ [1,8]. In particular,for λ,μ ∈ X+m(κ), it can be shown that the bundle A ⊗F[x±1] C̃(Lκ(μ),L†κ (λ)) is equivalent to the vector bundle ofconformal coinvariants (the dual of the bundle of conformal blocks) in conformal field theory (see [2, §7.3]).In [1], an action of the algebra H̃κ on the space C̃(N,M) was constructed through the Knizhnik–Zamolodchikovconnection, and some conjectures were proposed concerning the case where N and M are integrable, that is, N =Lκ(μ), M = L†κ(λ) for some λ,μ ∈ X+m(κ). Conjecture 5.4.2 in [1] states that C̃(Lκ(μ),L†κ (λ)) is isomorphic toL̃κ(λ,μ) when λ,μ ∈ X+m(κ) and λ − μ |= n, and Conjecture 6.2.6 in [1] gives a decomposition of the W -invariantpart C̃(Lκ(μ),L†κ (λ))W into the weight spaces with respect to the commutative subalgebra F[ ε∨]W .The irreducible H̃κ -modules L̃κ(λ,μ) (λ,μ ∈ X+m(κ)) are studied in [7] and their structures are described explic-itly in terms of tableaux on periodic skew diagrams. This combinatorial description is applied to prove these twoconjectures.Theorem 2.2. Conjecture 5.4.2 and Conjecture 6.2.6 in [1] are true.The details of the proof will be published in the forthcoming papers. In the rest of this note, we will presentthe analogous statements for the rational algebra Hκ . These statements for Hκ are derived from the correspondingstatements for H̃κ through the relation between the category of Hκ -modules and that of H̃κ -modules establishedin [6].3. The rational double affine Hecke algebra and character formulaLet κ ∈ F and m ∈ Z1. For a lowest weight ĝ-module M of level −κ + m, we setT (M) = V [x1] ⊗ . . . ⊗ V [xn] ⊗ M, C(M) = T (M)g[t].Put ekl[p] = ekl ⊗ tp ∈ ĝ for k, l ∈ [1,m], p ∈ Z, where ekl ∈ g denotes the matrix unit with only non-zero entries 1at (k, l)-th component. For i ∈ [1, n], defineωi =∑p∈Z1∑k,l∈[1,m]1⊗i−1 ⊗ ekl[p − 1] ⊗ 1⊗n−i ⊗ elk[−p],which is an element of some completion of U(ĝ)⊗n+1 and defines a well-defined operator on T (M).Let σij ∈ EndF(F[x ]) denote the permutation of xi and xj . Let πij ∈ EndF(V ⊗n) denote the permutation of i-thand j -th component of the tensor product. Note that T (M) ∼= F[x ] ⊗ V ⊗n ⊗ M as a space, through which we regardσij and πij as elements in EndF(T (M)). Define the operators on T (M) byDi = κ ∂∂xi+∑j∈[1,n], j =i1xi − xj (1 − σij )πij − ωi.A parallel argument as in [1, §4] implies the following:Theorem 3.1. Let M be a lowest weight ĝ-module of level −κ + m.(i) There exists a unique algebra homomorphism θ : Hκ → EndF(T (M)) such thatθ(ε∨i) = Di (i ∈ [1, n]), θ(εi) = xi (i ∈ [1, n]), θ(si) = πi i+1σi i+1 (i ∈ [1, n − 1]).(ii) The Hκ -action on T (M) above preserves the subspace g[t]T (M). Therefore, θ induces an Hκ -module structureon C(M).386 T. Suzuki / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 383–386Define the elements ui (i ∈ [1, n]) of Hκ by ui = εi ε∨i +∑i−1j=1 sji . It follows that u1, . . . , un are pairwisecommutative. Now, let κ ∈ Z1 and set Λ+κ (m,n) = {λ ∈ X+m(κ) | λ |= n, λm  1}. As a rational analogue of [1,Conjecture 5.4.2], we haveTheorem 3.2. Let λ ∈ Λ+κ (m,n). Then C(L†κ (λ)) ∼= Lκ(λ), and moreover it is semisimple over F[u ].Remark 3.3. It is shown in [6] that the assignment λ → Lκ(λ) induces a bijection between the set⊔m∈[1,κ] Λ+κ (m,n)and the set of isomorphism classes of finitely generated irreducible Hκ -modules which are semisimple over F[u ] andlocally nilpotent for the subalgebra∑i∈[1,n] ε∨i F[ ε∨]. In particular, any irreducible module of this class is realized asthe space C(L†κ (λ)) of coinvariants for some λ ∈ Λ+κ (m,n).For an Hκ -module M , put MW = {v ∈ M | wv = v ∀w ∈ W }, which we call the symmetric part (or the sphericalsubspace) of M , and regard as an F[u ]W -module. Recall that Hκ has the grading operator ∂ := κ−1∑ni=1 ui ∈ F[u ]W ,satisfying [∂, εi] = εi, [∂, ε∨i ] = −ε∨i and [∂,w] = 0. Analogously to [1, Corollary 6.2.6], a decomposition of Lκ(λ)W(λ ∈ Λ+κ (m,n)) into weight spaces with respect to the algebra F[u ]W is described explicitly. This gives a simple andremarkable formula for the trace TrLκ (λ)W q∂ , which we will describe in the sequel.Let λ be a partition of n and let T be a standard tableau on the Young diagram corresponding to λ; namely, T is abijection from the diagram λ (viewed as a subset of Z2) to the set [1, n] which is strictly increasing in both row andcolumn directions. Define λ(i)T (i ∈ [1, n]) as the element of X+m corresponding to the Young diagram T −1([1, i]) ⊂ Z2.For  ∈ Z1, a standard tableau T is called an -restricted standard tableau if λ(i)T ∈ X+m( + m) for all i ∈ [1, n]. LetSt()(λ) denote the set of -restricted standard tableaux on λ. For T ∈ St()(λ) and i ∈ [1, n], definehi(T ) ={1 if a < a′,0 if a  a′, where (a, b) = T−1(i), (a′, b′) = T −1(i + 1), (1)and defineǨ()λ (1n)(q) =∑T ∈St()(λ)q∑ni=1(n−i)hi (T ). (2)The polynomial given by K()λ (1n)(q) := Ǩ()λ′ (1n)(q) is called the -restricted Kostka polynomial associated with thepartitions λ and (1n), where λ′ is the conjugate of λ.Theorem 3.4. Let m,κ ∈ Z1 and λ ∈ Λ+κ (m,n) . ThenTrLκ (λ)W q∂ = q12κ∑mi=1 λi(λi−2i+1)(1 − q)(1 − q2) · · · (1 − qn)Ǩ(κ−m)λ(1n) (q).AcknowledgementsThe author would like to thank K. Iohara for his valuable comments on the draft of this paper.References[1] T. Arakawa, T. Suzuki, A. Tsuchiya, Degenerate double affine Hecke algebras and conformal field theory, in: M. Kashiwara, et al. (Eds.),Topological Field Theory, Primitive Forms and Related Topics, The Proceedings of the 38th Taniguchi Symposium, Birkhäuser, 1998, pp. 1–34.[2] B. Bakalov, A. Kirillov Jr., Lecture on Tensor Categories and Modular Functors, University Lecture Series, vol. 21, Amer. Math. Soc., 2001.[3] I.V. Cherednik, Special bases of irreducible representations of a degenerate affine Hecke algebra, Funct. Anal. Appl. 20 (1) (1986) 76–78.[4] C. Dunkl, E. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc. 86 (2003) 70–108.[5] A. Ram, Skew shape representations are irreducible, in: Combinatorial and Geometric Representation Theory (Seoul, 2001), in: Contemp.Math., vol. 325, Amer. Math. Soc., Providence, RI, 2003, pp. 161–189.[6] T. Suzuki, Rational and trigonometric degeneration of double affine Hecke algebras of type A, Int. Math. Res. Not. 37 (2005) 2249–2262.[7] T. Suzuki, M. Vazirani, Tableaux on periodic skew diagrams and irreducible representations of the degenerate double affine Hecke algebras oftype A, Int. Math. Res. Not. 27 (2005) 1621–1656.[8] M. Varagnolo, E. Vasserot, From double affine Hecke algebras to quantized affine Schur algebras, Int. Math. Res. Not. 26 (2004) 1299–1333.

Double affine Hecke algebras, conformal coinvariants and Kostka polynomials

Takeshi Suzuki

Comptes Rendus Mathématique

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Double affine Hecke algebras, conformal coinvariants and Kostka polynomials

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