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 irreducible∈ 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 ∈ 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
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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 { | ≤ ≤ } 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
(+) 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 ∈ Z
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vector of Lλ,then
Eav+ =0, 1 a v˜+Fa =0, 1 a 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+: ∈ k (2.1) V ⊗ = mλLλ, λ Λn(k) ∈M where 0 (2.2) xva = tba,xvb, x Uq(gl ), h i ∀ ∈ 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 ∈ Uq(gl ) , a, b. Under the co-multiplication of Uq(gl ) ,wehave n ∀ 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 { | } 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 ∈ 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. Thus (λ) Proposition 2.1. t 1 α, β dim Lλ,λ Λn forms a basis of q(gl ). { βα| ≤ ≤ ∈ } T n This will be called a Peter-Weyl basis of q(gl ). T n Below we give a more explicit description of the bialgebra q(gln)intermsof generators and relations following [FRT, Ta]. Let R be the RT-matrix associated with the irreducible Uq(gln)-representation π furnished by the module V in the standard basis: n 1 R := 1 1+ (q 1)eaa eaa +(q q− ) eab eba. ⊗ − ⊗ − ⊗ a=1 a (2.3) R(π π)∆(x)=(π π)∆0(x)R, x Uq(gl ), ⊗ ⊗ ∀ ∈ n where ∆0 is the opposite co-multiplication of Uq(gln). Definition 2.2 ([FRT]). Let Cq[Xab] be the unital associative algebra over C(t) generated by Xab,1 a, b n, subject to the following relations: ≤ ≤ (2.4) R12X1X2 = X2X1R12, where R12 = R 1,X1 = eab 1 Xab,X2 = 1 eab Xab. ⊗ a,b ⊗ ⊗ a,b ⊗ ⊗ As is well known, Cq[XabP] has the structure of a bialgebra,P with the comultipli- cation ∆0(Xab)= c Xac Xcb, and co-unit 0(Xab)=δab. Introduce an order > for⊗ the pairs (a, b), 1 a, b n, such that (a, b) > (a+k, c), P ≤ ≤(k) > kab (a, b + k) > (a, b), if k is a positive integer. Let X = a,b (Xab) ,wherethe product is arranged according to the order > of the indices of X’s in such a way Q n2 that Xab is positioned in front of Xcd if (a, b) > (c, d). The k Z+ appearing in (k) n ∈ the superscript of X denotes the square matrix (kab)a,b=1.Setk = ab kab. The following lemma is well known. It is an immediate consequence| of| the defining relations (2.4). P (k) n2 Lemma 2.1. The monomials X , k Z+ ,formabasisofCq[Xab]. ∈ The following result was established in [Ta]: Theorem 2.1 ([Ta]). The following map defines a bialgebra isomorphism: > (k) (k) kab (2.5) ı : Cq[Xab] q(gln),XT := (tab) . →T 7→ a,b Y License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HOWE DUALITY FOR QUANTUM GROUPS 2685 3. Multiplicity free actions Let Θ = 1, 2, ..., m , m n. We consider a number of Hopf subalgebras of { } ≤ Uq(gln) related to m.OneisUq(glm) generated by the elements of the following 1 set: = K± , 1 i m; Ej ,Fj , 1 j It is easy to show that q(gln) forms a left module algebra over Uq(gln) under the following left action: T Φ:Uq(gl ) q(gl ) q(gl ), n ⊗T n →T n x f f f ,x, ⊗ 7→ (1)h (2) i X(f) where we have used Sweedler’s notation for the comultiplication of f q(gln): ∆(f)= f f . Similarly, under the following right action: ∈T (f) (1) ⊗ (2) P Ψ:˜ q(gl ) Uq(gl ) q(gl ), T n ⊗ n →T n f x f ,xf , ⊗ 7→ h (1) i (2) X(f) q(gl ) forms a right module algebra over Uq(gl ). Closely related to Ψisthe˜ T n n following left Uq(gl )-action on q(gl ): n T n Ψ:Uq(gl ) q(gl ) q(gl ), n ⊗T n →T n x f f ,S(x) f . ⊗ 7→ h (1) i (2) X(f) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2686 R. B. ZHANG q(gl )formsaleftUq(gl )-module algebra under Ψ with respect to the opposite T n n comultiplication of Uq(gln). It is obvious but important to observe that the action Φ commutes with Ψ and Ψ.˜ Also these actions naturally restrict to actions of the subalgebras Uq(glm), Uq(p+)andUq(p )on q(gln). − T Below we shall use the following notation. For any f q(gl ),x Uq(gl ), ∈T n ∈ n Ψx(f):=Ψ(x f), Φx(f):=Φ(x f), Ψ˜ x(f):=Ψ(˜ f x). ⊗ ⊗ ⊗ Definition 3.2. [n] m := f q(gln) ΨK 1 (f)=f, c > m, Ψx(f)=0,x Uq(p )− . R { ∈T | c± ∈ − } [n] Lemma 3.1. The following set forms a basis of m : R > kib (tib) kib Z+ . | ∈ 1 i m,1 b n ≤ ≤ Y≤ ≤ (k) (k) [n] Proof. Observe that T = ı(X ) m if and only if it belongs to this set. Now ∈R the lemma follows from Theorem 2.1. [n] Theorem 3.1. (1) m forms a left module algebra over Uq(gln) under the action R ˜ Φ, and also forms a right module algebra over Uq(glm) under the action Ψ. [n] (2) Φ(Uq(gl )) and Ψ(U˜ q(gl )) are mutual centralizers in End ( m ).Fur- n m C(t) R thermore, with respect to the joint action Φ Ψ˜ of Uq(gl ) Uq(gl ), ⊗ n ⊗ n [n] (0) (3.1) = Lλ L˜ , Rm ∼ ⊗ λ λ Λm M∈ (0) where L˜ denotes the irreducible right Uq(gl )-module with highest weight λ Λm. λ m ∈ Proof. Consider part (1) first. Let 1 − := [C(t)(Ka 1) C(t)(Ka− 1)] Uq(p )−, − C a>m − ⊕ − ! M M [n] which forms a two-sided co-ideal of Uq(gl ). For any f,g m , n ∈R Ψx(fg)= Ψx (f)Ψx (g)=0, x −. (1) (2) ∀ ∈C X(x) [n] [n] Thus, fg belongs to m ,thatis, m forms a subalgebra of q(gln). R R [n] T Since the left actions Φ and Ψ commute, m clearly forms a left Uq(gln)-module R [n] under Φ. Consider Ψ˜ u(f)= f ,u f for u Uq(gl )andf m .Now h (1) i (2) ∈ m ∈R Ψ Ψ˜ (f)=Ψ˜ 1 (f)=Ψ˜ (f),c>m, Kc u P KcuKc− u ˜ ΨxΨu(f)=ΨxS 1(u)(f)=0,x Uq(p )−, − ∈ − 1 where in the last equation we have used the fact that xS− (u) Uq(p )−.Thus [n] ˜ ∈ − m forms a right Uq(glm)-module under the action Ψ. The Φ(Uq(gln)) and R˜ [n] Ψ(Uq(glm)) clearly preserve the algebraic structure of m in the sense of Defi- nition 3.1. This completes the proof of part (1). R (0) Now we prove the decomposition (3.1). This requires some preparation. Let Lλ be an irreducible finite-dimensional Uq(p )-module with highest weight λ.Then − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HOWE DUALITY FOR QUANTUM GROUPS 2687 (0) Lλ is irreducible with respect to Uq(glm) Uq(p ), and the elements of Uq(p )− (0) ⊂ − − act on Lλ by zero. Let (0) (3.2) λ := ζ Lλ q(gln) (id Ψp)ζ =(S(p) id)ζ, p Uq(p ) . O { ∈ ⊗T | ⊗ ⊗ ∀ ∈ − } Proposition 3.1. λ forms a left Uq(gl )-moduleundertheaction O n Uq(gl ) λ λ,xζ (id Φx)ζ, n ⊗O →O ⊗ 7→ ⊗ Lλ,λ Λn, and λ = ∈ O ∼ 0, otherwise. Remark. Proposition 3.1 is the quantum Borel-Weil theorem [APW, PW, GZ] in our context. It can be easily deduced [GZ] from the Peter-Weyl theory for q(gln), in particular, Proposition 2.1. T Now we turn back to the proof of (3.1). Consider the Φ ΨactionofUq(gln) [n] [n] ⊗ ⊗ Uq(gl )on m . One can always decompose m into m R R [n] (0) ∗ = mλL L , Rm ∼ µ(λ) ⊗ λ λ Λm M∈ (0) ∗ (0) where Lλ is the dual vector space of the left Uq(glm)-module Lλ ,which has a natural left Uq(glm)-module structure. The mλ Z+ is the multiplicity of ∈ (0) ∗ [n] the irreducible Uq(gl ) Uq(gl )-module L L in m ,andµ(λ) Λn n ⊗ m µ(λ) ⊗ λ R ∈ (0) depends on λ. Take an irreducible Uq(gl )-module Lν, ν Λm,whichcanbe m ∈ extended to a unique Uq(p )-module with Kc, c>m, acting by the identity map, − (0) and Ecc 1, c>m, acting by zero. We can therefore construct an ν from Lν .It is easy to− see that O U (gl ) (0) [n] q m ν = L , O ν ⊗Rm where the Uq(glm)-action is defined by x(v f)= (x) x(1)v Ψx(2) (f), for all (0) [n] ⊗ ⊗ x Uq(gl ), v f Lν m . The Borel-Weil Theorem (Proposition 3.1) forces ∈ m ⊗ ∈ ⊗R P mλ =1,µ(λ)=λ, λ Λm. ∀ ∈ Thus with respect to Φ(Uq(gl )) Ψ(Uq(gl )), n ⊗ m [n] (0) ∗ = Lλ L . Rm ∼ ⊗ λ λ Λm M∈ This is equivalent to (3.1) with respect to Φ(Uq(gl )) Ψ(U˜ q(gl )) since the an- n ⊗ m tipode S of Uq(gln) is invertible. ˜ The decomposition (3.1) implies that Φ(Uq(gln)) and Ψ(Uq(glm)) are mutual [n] centralizers in End ( m ). C(t) R The m = n case of the theorem leads to Corollary 3.1. As a Uq(gl ) Uq(gl )-module under the joint action Φ Ψ˜ , n ⊗ n ⊗ q(gl ) = Lλ L˜λ. T n ∼ ⊗ λ Λn M∈ This is also an immediate consequence of Proposition 2.1. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2688 R. B. ZHANG Definition 3.3. [n] + := f q(gl ) Φ 1 (f)=f, c > m, Φx(f)=0,x Uq(p+) . Lm { ∈T n | Kc± ∈ } [n] Similar to the discussions on m , we have the following results. R [n] Lemma 3.2. The following set forms a basis of m : L > kbi (tbi) kbi Z+ . | ∈ 1 i m,1 b n ≤ ≤ Y≤ ≤ [n] Theorem 3.2. (1) m forms a left module algebra over Uq(glm) under the action L ˜ Φ, and also forms a right module algebra over Uq(gln) under the action Ψ. (2) With respect to the joint action Φ Ψ˜ of Uq(gl ) Uq(gl ), ⊗ m ⊗ n [n] (0) (3.3) = L L˜λ. Lm ∼ λ ⊗ λ Λm M∈ The theorem can be proven in a way similar to the proof of Theorem 3.1. The (0) following result will be of crucial importance. An irreducible Uq(glm)-module Lλ + with highest weight λ extends to a unique Uq(p+)-module with Uq(p+) acting by ˜ (0) zero. Let λ denote the lowest weight of Lλ . Define (0) (3.4) ˜λ := ζ L q(gl ) (id Φp)ζ =(S(p) id)ζ, p Uq(p+) . O { ∈ λ ⊗T n | ⊗ ⊗ ∀ ∈ } Proposition 3.2. ˜λ forms a right Uq(gl )-moduleundertheaction O n ˜λ Uq(gl ) ˜λ,xζ (id Ψ˜ x)ζ, O ⊗ n → O ⊗ 7→ ⊗ ˜ ˜ ˜ L λ˜, λ Λn, and λ = − ∈ O ∼ 0,− otherwise. Remark. This is a variant of the quantum Borel-Weil theorem [APW, PW, GZ] and easily follows [GZ] from Proposition 2.1. Definition 3.4. [n] m ,m n, (3.5) m,n := R[m] ≤ V ( n ,m n. L ≥ Combining Theorems 3.1 and 3.2 we arrive at the quantum Howe duality, The- orem 1.1. 4. Howe duality implies Schur duality We adapt the definition of the q-Weyl group of [KR] to the Jimbo setting of quantized enveloping algebras. Denote by the set of finite-dimensional Uq(gl )- M n modules of type I. Every M is semisimple, i.e., M = Lλ Mλ. Let Aut( ) ∈M ⊗ M be the set consisting of the automorphisms of the underlying C(t)-vector spaces of all the elements of .Let be the set of the maps σ : L Aut( ) such that σ(M):M M, andM are ofG the form M→ M → (4.1) σ(M)= σ(Lλ) idM , ⊗ λ X License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HOWE DUALITY FOR QUANTUM GROUPS 2689 for any M = Lλ Mλ . Therefore, a σ is uniquely determined once ⊗ ∈M ∈G σ(Lλ) are given for all Lλ . Clearly, the diagram L ∈M σ(M) M M −→ α α ↓↓ σ(M 0) M 0 M 0 −→ commutes for any morphism α : M M 0 of finite-dimensional Uq(gl )-modules → n of type I. Furthermore, forms a group, with the multiplication (σ, σ0) σσ0 G1 7→ and inverse map σ σ− , respectively, defined by (σσ0)(M)=σ(M)σ0(M), and 1 7→1 σ− (M)=(σ(M))− , for all M . The identity of this group is the map ∈M M idM , M.Theq-Weyl group q(n)ofUq(gln) is a subgroup of which we now7→ define.∀ W G (j,c) Let U be the type I irreducible Uq(gl2)-module with the highest weight (j,c) (j + c)1 +(j c)2 Z1 Z2 such that j Z+/2. Consider a basis um m = j, j 1, ...,−1 j, ∈ j of⊕U (j,c) defined by ∈ { | − − − } (j,c) c+m (j,c) K1um = q um , (j,c) c m (j,c) K2um = q − um , (j,c) (j,c) E1u =[j + m +1][j m]u , m − m+1 (j,c) (j,c) (4.2) F1um = um 1, − j j q q− 2 (j,c) where [j]= q −q 1 . (Recall that q = t .) Define an automorphism w of the − − underlying vector space of U (j,c) by (j,c) (j,c) j m 2j(j+1) 2c2 [j m] (j,c) (4.3) w um =(1) − t− − − u m . − [j + m] − 1 Remark. Let = C[t, t− ] be the subring of C(t) consisting of the Laurent polyno- A mials in t.Thenw(j,c) defines an automorphism of the free -module U (j,c) with (j,c) A A basis um . { } a 1 1 Let Uq (gl2) be the subalgebra of Uq(gln) generated by Ea, Fa, Ka± and Ka±+1, a for a fixed a Remark. A q-Weyl group can also be defined with respect to right Uq(gln)-modules in the obvious way. Results parallel to Theorem 4.1 and Proposition 4.1 below hold for the right module version of the q-Weyl group. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2690 R. B. ZHANG For a finite dimensional type I Uq(gln)-module M, we define (4.4) (M)0 := v M (Ka q)v =0, a , { ∈ | − ∀ } and call it the space of zero weight vectors of M. Clearly (M)0 is stable under all the wa(M), thus forms a q(n)-submodule of M. Since each finite-dimensional W Uq(gln)-module of type I is isomorphic to the tensor product of a one-dimensional Uq(gln)-module with an Lλ, λ Λn, the study of the q(n)-modules (M)0 es- ∈k W sentially reduces to studying (V ⊗ )0, k Z+,whereV is the contravariant vector k ∈ Uq(gln)-module. Note that (V ⊗ )0 = 0 unless k = n. n The C(t)-space (V ⊗ ) has a basis vσ(1) vσ(2) ... vσ(n) σ Sn ,where 0 { ⊗ ⊗ ⊗ | ∈ } Sn is the symmetric group on n letters. Denote the action of the q-Weyl group on n n n (V ⊗ ) by τ : q(n) (V ⊗ ) (V ⊗ ) . 0 W × 0 → 0 Theorem 4.1. C(t)τ( q(n)) is a homomorphic image of the Hecke algebra a(n). More explicitly, W H τ(wa)τ(wb)=τ(wb)τ(wa), a b > 1, | − | (4.5) τ(wa)τ(wa+1)τ(wa)=τ(wa+1)τ(wa)τ(wa+1), 2 2 1 (4.6) (q τ(wa) q)(q τ(wa)+q− )=0, a. − ∀ Proof. It can be easily deduced from results of [KR] (or from the formulae (4.7) below) that the wa obey the relations of the braid group of type A.Thusweonly n need to prove the quadratic relations (4.6). Fix an index a;then(V ⊗ )0 is spanned a (1,1) (0,1) by the zero weight vectors of copies of the Uq (gl2)-modules U and U .Acting (1,1) (0,1) 3 1 on U 0 and U 0, wa takes eigenvalues q− and q− respectively. Hence the quadratic relations (4.6). − n We can describe the action of q(n)on(V ⊗ )0 more explicitly. Consider vectors n W ζa, ηa (V ⊗ ) of the form ζa := X va Y va+1 Z and ηa := X va+1 ∈ 0 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ Y va Z.Wehave ⊗ ⊗ 2 1 q τ(wa)(ζa)=(q q− )ζa ηa, 2 − − (4.7) q τ(wa)(ηa)= ζa. − Remark. Using the formulae (4.7) one can easily prove Theorem 4.1 by direct com- putations. n When q is specialized to 1, the action of q(n)on(V ⊗ )0 reduces to that of W the symmetric group Sn. To explain this in more precise terms, we consider the 1 free module V over ( = C[t, t− ]) generated by the elements of the standard A A A n basis va a =1, 2, ..., n of V .DenotebyV ⊗ the n-th power of V over .Set n { | n } A A A (V ⊗ )0 := w V ⊗ (Ka q)w =0, a ,whichisafree -module. Because of the A { ∈ A | − ∀n } A remark following equation (4.3), (V ⊗ )0 is invariant under q(n). Introduce to C A W the following -module structure: ψ : C C C, a(t) 1 a(1). We define A n A⊗ → ⊗ 7→ q(n) C and (V ⊗ )0 C using ψ, and consider the action of q(n) C AW n⊗A A ⊗A AW ⊗A on (V ⊗ )0 C.Letik, k =1, 2, ..., n, be distinct elements of 1, 2, ..., n .We have A ⊗A { } wa 1:vi vi ... vi 1 v v ... v 1, − ⊗ 1 ⊗ 2 ⊗ ⊗ n ⊗ 7→ sa(i1) ⊗ sa(i2) ⊗ ⊗ sa(in) ⊗ where sa Sn permutes a and a+1 while leaving the other elements of 1, 2, ..., n unchanged.∈ { } License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HOWE DUALITY FOR QUANTUM GROUPS 2691 Proposition 4.1. Assume that Lλ is an irreducible Uq(gln)-module of type I with highest weight λ Λn(n). Then the space of zero weight vectors (Lλ)0 forms the ∈ irreducible q(n)-module determined by the partition corresponding to λ.Also, H every irreducible q(n)-module is isomorphic to an (Lλ)0 with λ Λn(n). H ∈ Proof. Since any type I irreducible Uq(gln)-module Lλ with highest weight λ n ∈ Λn(n) can be embedded into V ⊗ , and this is also an embedding of C(t) q(n)- W modules, we only need to consider the case when Lλ is an irreducible Uq(gln)- n n n submodule of V ⊗ .LetLλ, := V ⊗ Lλ,and(Lλ, )0 := V ⊗ (Lλ)0,where n A n A ∩ A A ∩ V ⊗ is regarded as a subset of V ⊗ . Now we specialize t to 1 by using ψ. It follows fromA results of Lusztig and Rosso (see [CP]) on quantized enveloping algebras at generic q that 0 Lλ, C = Lλ, A A ∼ ⊗ 0 (Lλ, )0 C = (Lλ)0, A ⊗A ∼ 0 0 where Lλ is the irreducible gln-module with highest weight λ,and(Lλ)0 is the space 0 0 of zero weight vectors of Lλ.Itisknownthat(Lλ)0 forms the irreducible Sn-module characterized by the partition corresponding to λ. (This result is due to Schur. Modern treatments of it can be found in, e.g., [Ho, Ko].) Since dimC(t)(Lλ)0 = 0 dimC(Lλ)0,andany q(n)-submodule of (Lλ)0 would give rise to an Sn-submodule 0 H of (L )0, we conclude that (Lλ)0 is irreducible with respect to q(n). By recalling λ H thefact(see[Ma])that q(n)andC(t)Sn are isomorphic as associative algebras H and are semisimple, we easily see that (Lλ)0 is the irreducible left q(n)-module H Sλ determined by the partition corresponding to λ.TheSλ, λ Λn(n), exhaust ∀ ∈ all the irreducible q(n)-modules up to isomorphisms. H We now derive from Theorem 1.1 Jimbo’s quantum Schur duality at generic q [Ji], which amounts to the following statement. Theorem 4.2. Let V be the contravariant vector module over Uq(gln).Then m (4.8) V ⊗ = Lλ M˜ λ, ⊗ λ Λn (m) ∈M where M˜ λ is the irreducible right q(m)-module associated with λ. H Proof. We shall follow the same strategy as that of section 2.4.5.2 in [Ho]. Consider the space ( n,m) of zero Uq(gl )-weight vectors of n,m under the right V 0,Uq (glm) m V action of U (gl ). As a left U (gl )-module, ( ) is isomorphic to V m. q m q n n,m 0,Uq (glm) ⊗ Following Theorem 1.1, V m [n] [m] V ⊗ = L (L˜ )0, λ ⊗ λ λ Λn (m) ∈M [m] wherewehaveusedthefactthat(L˜ )0 =0ifandonlyif λ = m.Theright λ 6 | | module version of Proposition 4.1 immediately leads to what we seek to prove. 5. Some remarks Our proof of Theorem 1.1 makes essential use of the Peter-Weyl basis (Proposi- tion 2.1) of q(gl ) via Propositions 3.1 and 3.2. When q is specialized to a root of T n unity, Proposition 2.1 fails completely. However, q(gl ) is still well defined [PW], T n and so is also m,n. It is clearly true that m,n forms a module algebra under the V V License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2692 R. B. ZHANG ˜ left action Φ of Uq(glm) and right action Ψ of Uq(gln), with the actions of the two quantum algebras commuting. Then a natural question is whether the following relations hold at roots of unity: Φ(Uq(glm)) m,n =EndUq(gl )( m,n), |V n V ˜ Ψ(Uq(gln)) m,n =EndUq(gl )( m,n). |V m V It is quite likely that the answer is affirmative in view of the fact [PW] that the m m actions of Uq(gln)and q(m)onV ⊗ are centralizers of each other in EndC(V⊗ ) even at roots of unity. H References [APW] Andersen, H. H.; Polo, P.; Wen, K. X., Representations of quantum algebras.Invent. Math. 104 (1991) 1–59. MR 92e:17011 [CP] Chari, V.; Pressley, A., A guide to quantum groups. Cambridge University Press, Cam- bridge, 1994. MR 95j:17010 [GZ] Gover, A. R.; Zhang, R. B., Geometry of quantum homogeneous vector bundles and representation theory of quantum groups. I.Rev.Math.Phys.11 (1999) 533–552. MR 2000j:81108 [Ho] Howe, R., Perspectives on invariant theory. The Schur Lectures (1992). Eds. I. 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