PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 9, Pages 2681{2692 S 0002-9939(02)06892-2 Article electronically published on December 30, 2002 HOWE DUALITY AND THE QUANTUM GENERAL LINEAR GROUP R. B. ZHANG (Communicated by Dan M. Barbasch) Abstract. A Howe duality is established for a pair of quantized enveloping algebras of general linear algebras. It is also shown that this quantum Howe duality implies Jimbo's duality between Uq(gln) and the Hecke algebra. 1. Introduction A treatment of classical invariant theory based on the highest weight theory of representations of complex reductive algebraic groups was given in [Ho], where the organizing principle is the multiplicity free actions of reductive groups on algebraic varieties commonly referred to as Howe dualities. In the context of the general linear group over the complex field, the Howe duality amounts to the following result (or a variant of it in terms of right modules): Regard Ck as the natural (left) GLk(C)-module for each k.GLm(C) GLn(C) acts on the algebra of regu- × lar functions (Cm Cn)onCm Cn in a multiplicity free manner. Moreover, P ⊗ [m];⊗0 [n];0 [m];0 [n];0 (Cm Cn)= L L ,whereL (respectively L )de- P ⊗ λ Λmin(m;n) λ ⊗ λ λ λ notes the irreducible2 GL ( )-module (respectively GL ( )-module) with highest L m C n C weight λ,andΛmin(m;n) is the set of the highest weights corresponding to partitions of depth min(m; n). While this result can be proven by relatively elementary means, its≤ implications are far reaching. As shown in [Ho], the First and Second Fundamental Theorems of the invariant theory for the general linear group are all immediate consequences of this result. In particular, the celebrated Schur duality can be easily derived from the Howe duality. This paper aims to develop a quantum version of the Howe duality for a pair of quantized enveloping algebras of general linear algebras at generic q.Themain m n problem is to construct a non-commutative analogue m;n of (C C ). We achieve this by investigating the Hopf algebra of functionsV on theP quantum⊗ general linear group. With the m;n given in Definition 3.4, we prove V Theorem 1.1. m;n forms a module algebra under the left action of Uq(gl ) and V m right action of Uq(gln), with the actions of the two quantum algebras being mutual centralizers. Furthermore, [m] [n] m;n = L L~ ; V λ ⊗ λ λ Λ 2 minM(m;n) Received by the editors June 24, 2001 and, in revised form, April 7, 2002. 2000 Mathematics Subject Classification. Primary 17B37, 20G42, 17B10. c 2002 American Mathematical Society 2681 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2682 R. B. ZHANG [m] ~[n] where Lλ (respectively Lλ ) denote the irreducible left Uq(glm)-module (respec- tively right Uq(gln)-module) of type I [CP] with highest weight λ. The notion of a module algebra over a bi-algebra is explained in Definition 3.1. It is an old result of Jimbo's [Ji] dating back to the mid-1980s (and having since been treated by many other people) that there existed a duality between Uq(gln) and the Hecke algebra analogous to the celebrated Schur duality between GLn(C) and the symmetric group. We shall derive the quantum Schur duality from the quantum Howe duality established here. For doing this, we investigate the representations of the q-Weyl group [KR] of Uq(gln) on the spaces of zero weight vectors (see equation (4.4)) of finite-dimensional Uq(gln)-modules. They give rise to representations of the Hecke algebra, which we believe are of independent interest, thus are studied in some detail. 2. The quantum general linear group Let h∗ be the complex vector space which has a basis a 1 a n and is f j ≤ ≤ g endowed with a non-degenerate symmetric bilinear form ( ; ):h∗ h∗ C defined by ( ,)=δ .Seth = : The quantized enveloping algebra× ! U (gl )of a b ab Z∗ a Z a q n the general linear algebra gln is a Hopf algebra over the field of rational functions L 1 C(t)int. As a unital associative algebra, Uq(gln) is generated by Ka;Ka− ; 1 ≤ a n,andEb;Fb; 1 b<n, subject to the standard relations, which we spell out≤ in order to fix our notation:≤ 1 1 1 1 1 KaKa− =1;Ka± Kb± = Kb± Ka± ; 1 (b b+1,a) 1 (b b+1,a) KaEbKa− = q − Eb;KaFbKa− = q− − Fb; 1 1 1 EaFb FbEa = δab(KaK− K− Ka+1)=(q q− ); − a+1 − a − EaEb = EbEa;FaFb = FbFa; a b > 1; j − j (+) ( ) aa 1 = aa− 1 =0; S ± S ± where q = t2,and (+) 2 1 2 aa 1 =(Ea) Ea 1 (q + q− )Ea Ea 1 Ea + Ea 1 (Ea) ; S ± ± − ± ± ( ) 2 1 2 aa− 1 =(Fa) Fa 1 (q + q− )Fa Fa 1 Fa + Fa 1 (Fa) : S ± ± − ± ± We will also need the explicit form of the other structural maps of Uq(gln): the comultiplication ∆ : Uq(gl ) Uq(gl ) Uq(gl ), n ! n ⊗ n 1 ∆(Ea)=Ea KaK− +1 Ea; ⊗ a+1 ⊗ 1 ∆(Fa)=Fa 1+Ka− Ka+1 Fa; 1 ⊗1 1 ⊗ ∆(K± )=K± K± ; a a ⊗ a the co-unit :Uq(gln) C(t), ! 1 (Ea)=(Fa)=0,(Kb± )=1; and the antipode S :Uq(gl ) Uq(gl ), n ! n 1 1 1 1 S(Ea)= EaK− Ka+1;S(Fa)= KaK− Fa;S(K± )=K∓ : − a − a+1 a a We shall denote by Lλ the irreducible left Uq(gln)-module of type I (in the sense of [CP]) with highest weight λ h∗ . More explicitly, if v+ is the highest weight 2 Z License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HOWE DUALITY FOR QUANTUM GROUPS 2683 vector of Lλ,then Eav+ =0; 1 a<n; ≤ (a,λ) Kav+ = q v+; a: 8 ~ Similarly, we denote by Lλ, λ h∗ , the irreducible right Uq(gl )-module which 2 Z n admits a unique 1-dimensional subspace C(t)~v+ such that v~+Fa =0; 1 a<n; ≤ (a,λ) v~+Ka = q v~+; a: 8 ~ We shall call Lλ the irreducible right Uq(gln)-module of type I with highest weight λ. n k Define Λn := a=1 λaa λa Z+,λa λa+1 : Let Λk =Λn i=1 Zi f j 2 n ≥ g \ n if and only if k n.Forλ = λaa Λn,weset λ = λa.Let ≤P a=1 2 j j a=1L Λn(d)= λ Λn λ = d : The irreducible left (respectively right) Uq(gl )-module f 2 jj j g P P n Lλ (respectively L~λ) is finite dimensional if λ Λn. 2 Of particular interest for us here is the contravariant vector module V = L1 .It has the standard basis va 1 a n with the action of Uq(gl )givenby f j ≤ ≤ g n Eavb = δba+1va; Favb = δbava+1; Kavb =[1+(q 1)δab] vb: − We shall denote by π the irreducible Uq(gln)-representation relative to this basis. k The tensor power V ⊗ of V is completely reducible for any k Z+: 2 k (2.1) V ⊗ = mλLλ; λ Λn(k) 2M where 0 <mλ Z+ is the multiplicity of Lλ. 02 Let Uq(gl ) := f Uq(gl )∗ kerf contains a cofinite ideal of Uq(gl ) de- n f 2 n j n g note the finite dual of the quantized enveloping algebra Uq(gln). Standard Hopf 0 algebra theory asserts [Mon] that Uq(gln) has the structure of a Hopf algebra in- duced by the Hopf algebra structure of Uq(gln). Denote by m0,∆0, 0,andS0 the 0 multiplication, comultiplication, co-unit and antipode of Uq(gln) respectively. Let tab,1 a; b n, be the matrix elements of the contravariant vector repre- ≤ ≤ sentation of Uq(gln) relative to the standard basis. That is, the tab are elements of the dual space Uq(gln)∗ of Uq(gln) defined by (2.2) xva = tba;xvb; x Uq(gl ); h i 8 2 n b X where ; represents dual space pairing. Clearly the matrix elements of any h· ·i 0 finite-dimensional representation of Uq(gln)belongtoUq(gln) .Inparticular,tab 0 0 2 Uq(gl ) , a; b. Under the co-multiplication of Uq(gl ) ,wehave n 8 n ∆0(tab)= tac tcb: c ⊗ X 0 Definition 2.1. Denote by q(gl ) the subbialgebra of Uq(gl ) generated by the T n n tab,1 a; b n. ≤ ≤ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2684 R. B. ZHANG 0 Any finite-dimensional left Uq(gln)-module has a natural right Uq(gln) -co- (λ) module structure. In particular, if wα α =1; 2; :::; dimLλ is any basis of the f j g 0 finite-dimensional irreducible Uq(gln)-module Lλ, then the right Uq(gln) -comodule 0 (λ) (λ) (λ) (λ) action Lλ Lλ Uq(gl ) is given by wα w t ,wherethet ! ⊗ n 7! β β ⊗ βα βα are the matrix elements of the Uq(gln)-representation associated to Lλ relative to P (λ) the given basis. If λ Λn, then it follows from (2.1) that all the t belong to 2 αβ q(gl ). Furthermore, Burnside's theorem implies that the matrix elements of all T n the irreducible representations of Uq(gln) with highest weights in Λn are linearly independent.