GEOMETRIC SCHUR DUALITY OF TWO PARAMETER QUANTUM GROUP OF TYPE A HAITAO MA, ZONGZHU LIN, AND ZHU-JUN ZHENG 7 1 0 Abstract. In this paper, we give an geometric description of the Schur-Weyl duality for 2 two-parameter quantum algebras U (gl ), where U (gl ) is the deformation of U (I, ), v,t n v,t n v · n the classic Shur-Weyl duality (U (gl ),V⊗d,H (r,s)) can be seen as a corollary of the r,s n d a Shur-Weyl duality (U (gl ),V⊗d,H (v,t)) by using the galois descend approach. we also J v,t n d 2 establish the Shur-Weyl duality between the algebras Uv,^t(glN)m, Uv,\t(glN)m and Heck 2 algebra Hk(v,t). ] A Q . 1. Introduction h t a Schur-Weyl duality is a classical method to construct irreducible modules of simple Lie m groups out of the fundamental representations [W46], The quantum version for the quantum [ enveloping algebra U (sl ) and the Hecke algebra H (S ) has been one of the pioneering q n q m 1 examples [13] in the fervent development of quantum groups. Two-parameter general linear v and special linear quantum groups [21, 8, 4] are certain generalization of the one-parameter 4 1 Drinfeld-Jimbo quantum groups [7, 12]. The two-parameter quantum groups also had their 1 origin in the quantum inverse scattering method [20] as well as other approaches [14, 6]. 6 So far, lots of mathematicians had studied the quantum groups and two parameter quan- 0 . tum group. For example, geometric Shur-Jimbo duality of type A was studied by Beilinson, 1 0 Lusztig and Mcpherson [BLM90]. And the Shur-like duality of type B/C and D were dis- 7 covered by Bao-Wang [BKLW14]and Fan-Li[FL14]. 1 : Especially, Fan and Li had found another version of two parameter quantum group by v the way of perverse sheaves [FL13]. But the question how the two parameter quantum i X group U (gl ) can be seen as the deformation of U (gl ) didn’t solve in their work. So it is v,t n v n r a necessary for us to give the new graded structure on Uv(gln) such that Uv,t(gln) can be seen as the deformation of U (gl ). v n Fan and Li found two new quantum group U and Um, and gave the Shur-Weyl duality between them and the Iwahori-hecke algebra of type D [FL14]. In our following paper, d similartotheFanandLi’swork, wewillgivetwonewtwoparameterquantumgroupU and v,t Um. We can also give the Shur-Weyl duality between them and the two parameter Iwahori- v,t hecke algebra of type D through the geometric way. In order to give the comutiplication of d the two new two parameter quantum group U , Um and use the comutiplication structure v,t v,t to give the Shur-Weyl duality algebraically. That is, ^ ∆ : U U (gl )m U, v,t N → ⊗ 1991 Mathematics Subject Classification. 17B37,14L35, 20G43. Key words and phrases. Iwahori-Hecke algebra of type A, flag variety of type A, Schur-type duality. 1 2 HAITAOMA, ZONGZHULIN,ANDZHU-JUNZHENG ∆ : Um U \(gl )m Um. v,t N → ⊗ ^ \ So it is reasonable for us to give structure of the new quantum group U (gl )m, U (gl )m v,t N v,t N and the Shur-Weyl duality between them and H (d). v,t In this work, at first, we give a new version of two parameter quantum group U (gl ), v,t n which is the deformation of U (gl ) similar to the approach appear in [FL13]. Second, we v n ^ would like to give the geometric realization of three quantum groups U (gl ), U (gl )m, v,t n v,t N \ U (gl )m. At the same time, we also give the Shur-Weyl duality between algebras U (gl ), v,t N v,t n ^ \ U (gl )m, U (gl )m and the Hecke algebra H (d). Since the classical two parameter v,t N v,t N v,t quantum group U (gl ) is the subalgebra of the new version U (gl ). we would like to r,s n v,t n use the Galois descend approach to understand the two different versions of two parameter quantum groups. The classical Shur-Weyl duality (U (gl ),V⊗d,H (r,s)) can be seen as r,s n d a corollary of the Shur-Weyl duality (U (gl ),V⊗d,H (v,t)) by using the galois descend v,t n d theory. That is, there exist a Galois group G such that (U (gl )G,(V⊗d)G,H (v,t)G) is also v,t n d Shur-Weyl duality, and U (gl )G = (U (gl ), H (v,t)G = H (r,s). v,t n ∼ r,s n d ∼ d 2. Deformation 1 0 0 ... 0 0 1 1 0 ... 0 0  −  0 1 1 ... 0 0 2.1. The algebra U (gl ). let Ω = . −. . . . . is associated to the v,t n  .. .. .. .. .. ..     0 0 0 ... 1 0     0 0 0 ... 1 1   − n×n cartan matrix of type A . Let I = 1,2, ,n . To Ω, we associate the following three n { ··· } bilinear forms on ZI. (1) i,j = Ω , i,j I. ij h i ∀ ∈ (2) i,j = 2δ Ω Ω , i,j I. ij ii ij − ∀ ∈ (3) i j = i,j + j,i , i,j I. (cid:2) (cid:3) · h i h i ∀ ∈ Definition 2.1.1. The two-parameter quantum algebra U (gl ) associated to A is an v,t n n−1 associative Q(v,t)-algebra with 1 generated by symbols E ,F , i I A±1,B±1, i I′ = i i ∀ ∈ j j ∀ ∈ GEOMETRIC SCHUR DUALITY OF TWO PARAMETER QUANTUM GROUP OF TYPE A 3 I n and subject to the following relations. ∪{ } (R1) A±1A±1 = A±1A±1, B±1B±1 = B±1B±1, i j j i i j j i A±1B±1 = B±1A±1, A±1A∓1 = 1 = B±1B∓1. i j j i i i i i (R2) A E A−1 = vhi,jithi,jiE , B E B−1 = v−hi,jithi,jiE , i j i j i j i j A F A−1 = v−hi,jit−hj,iiF , B F B−1 = vhi,jit−hj,iiF . i j i j i j i j A B B A i i+1 i i+1 (R3) E F F E = δ − . i j − j i ij v v−1 − (R4) (−1)pt−p(p′−2hii,·jii+2hji·,iii)Ei(p′)EjEi(p) = 0, if i 6= j, p+p′X=1−2ii··ji (−1)pt−p(p′−2hii,·jii+2hji·,iii)Fi(p)FjFi(p′) = 0, if i 6= j, p+p′X=1−2ii··ji where E(p) = Eip . j,n = 0, n,j = −1 if j = n−1; , j I. i [p]!vi,ti h i h i (cid:26) 0 else . ∈ The algebra U (gl ) has a Hopf algebra structure with the comultiplication ∆, the counit v,t n ε and the antipode S given as follows. ∆(A±1) = A±1 A±1, ∆(B±1) = B±1 B±1, i i ⊗ i i i ⊗ i ∆(E ) = E A B +1 E , ∆(F ) = F 1+B A F , i i i i+1 i i i i i+1 i ⊗ ⊗ ⊗ ⊗ ε(A±1) = ε(B′±1) = 1, ε(E ) = ε(F ) = 0, S(A±1) = A∓1, i i i i i i S(B±1) = B∓1, S(E ) = E B A , S(F ) = A B F . i i i − i i i+1 i − i i+1 i The algebra U (gl ) admits a ZI′ ZI′-grading by defining the degrees of generators as v,t n × follows. deg(E ) = (i,0), deg(F ) = (0,i), i i n n ( ( 1)kk, ( 1)kk) if j is even, − − deg(A ) = deg(B ) =  k=j k=j j j Pn P  ( ( 1)k+1k, n ( 1)k+1k) if j is odd.  − k=j − k=j P P We can define a bilinear form on ZI′ ZI′ by  × [γ,η]′ = [γ ,η ] [γ ,η ] 2 2 1 1 − for any γ = (γ ,γ ),η = (η ,η ) ZI′ ZI′. Then on U (gl ), we can define a new 1 2 1 2 v,t n ∈ × multiplication ′′ ” by ∗ (4) x y = t−[|x|, |y|]′xy, ∗ for any homogenous elements x,y U (gl ). Since [,]′ is a bilinear form, (U (gl ), ) v,t n v,t n ∈ ∗ is an associative algebra over Q(v,t). We define a multiplication, denoted by “ ”, on ∗ 4 HAITAOMA, ZONGZHULIN,ANDZHU-JUNZHENG U (gl ) U (gl ) by v,t n v,t n ⊗ (5) (x y) (x′ y′) = x x′ y y′. ⊗ ∗ ⊗ ∗ ⊗ ∗ This gives a new algebra structure on U (gl ) U (gl ). (U (gl ), ) has a Hopf algebra v,t n v,t n v,t n ⊗ ∗ structure with the comultiplication ∆∗, the counit ε∗ and the antipode S∗. The image of generators E ,F ,A and B−1 under the map ∆∗ (resp. ε∗ and S∗) are the same as the ones i i i i under the map ∆ (resp. ε and S) defined above. Lemma 2.1.2. Under the new multiplication “ ”, the defining relations of U (gl ) can be v,t n ∗ rewritten as follows. (R∗1) A±1 A±1 = A±1 A±1, B±1 B±1 = B±1 B±1, i ∗ j j ∗ i i ∗ j j ∗ i A±1 B±1 = B±1 A±1, A±1 A∓1 = 1 = B±1 B∓1. i ∗ j j ∗ i i ∗ i i ∗ i (R∗2) A E A−1 = vhi,jiE , B E B−1 = v−hi,jiE , i ∗ j ∗ i j i ∗ j ∗ i j A F A−1 = v−hi,jiF , B F B−1 = vhi,jiF . i ∗ j ∗ i j i ∗ j ∗ i j A B B A (R∗3) E F F E = δ i ∗ i+1 − i ∗ i+1, i,j I. i ∗ j − j ∗ i ij v v−1 ∀ ∈ − 1 a (R∗4) ( 1)p − ij E∗p E E∗p′ = 0, if i = j, − p i ∗ j ∗ i 6 p+pX′=1−aij (cid:20) (cid:21)v 1 a ( 1)p − ij F∗p F F∗p′ = 0 if i = j, − p i ∗ j ∗ i 6 p+pX′=1−aij (cid:20) (cid:21)v where a = 2i·j and E∗p = E E E for p copies. We notice that these relations are ij i·i i i ∗ i ∗···∗ i the specialization of (R1)-(R4) at t = 1. Proof. The relation R3, R4 agrees with the one in [FL13, 4. 2], whose proof is also the same as the one for type-A case. Next we show R∗2. A E = t−[|Ai|, |Ej|]′A E = t−[|Ai|, |Ej|]′vhi,jithi,jiE A = t[|Ej|,kAi|]′−[|Ai|, |Ej|]′vhi,jithi,jiE A i j i j j i j i ∗ ∗ and [ E , A ]′ [ A , E ]′ = ( j,i i,j ) ( j,i+1 i+1,j )+ +( 1)n−i( j,n n,j ). j i i j | | k | − | | | | h i−h i − h i−h i ··· − h i−h i = i+1,j = 1 = i,j if i = j, h i −h i = ( j,i i,j ) ( j,i+1 i+1,j ) = 1 = i,j if i = j +1,  h i−h i − h i−h i −h i [ E , A ]′ [ A , E ] =  = 0 = i,j if i j > 1, j i i j  | | k | − | | | |  −h i −    = j,i i+1,j = 0 = i,j if j = i+1, h i−h i −h i  = j 1,j + j,j 1 = 0 = i,j if j i > 1.  −h − i h − i −h i −   Therefore,  A E = vhi,jiE A . i j j i ∗ ∗ (cid:3) All other identity in R2 can be shown similarly. GEOMETRIC SCHUR DUALITY OF TWO PARAMETER QUANTUM GROUP OF TYPE A 5 Theone-parameterquantumalgebraU (I, )associatedto(I, )isdefinedastheassociative v · · Q(v)-algebra with 1 generated by symbols E ,F ,A±1, B±1, i I and subject to relations i i i i ∀ ∈ (R*1)-(R*4). U (I, ) has a Hopf algebra structure with the comultiplication ∆ , the counit v 1 · ε and the antipode S . The image of generators E ,F ,A ,B under the map ∆ (resp. ε 1 1 i i a a 1 1 and S ) are the same as the ones under the map ∆ (resp. ε and S) defined above. 1 Let U (I, ) := U (I, ) Q(v,t). The Hopf algebra structure on U (I, ) can be v,t v Q(v) v · · ⊗ · naturally extended to U (I, ). From the above analysis, we have the following theorem. v,t · Theorem 2.1.3. If (I, ) is the Cartan datum associated to Ω , then there is a Hopf-algebra n · isomorphism (U (gl ), ,∆∗,ε∗,S∗) (U (I, ), ,∆ ,ε ,S ), v,t n v,t 1 1 1 ∗ ≃ · · sending the generators in U to the respective generators in U (I, ). v,t v,t · 3. A geometric setting 3.1. Preliminary. Let F be a finite field of q elements and of odd characteristic. d is a q fixed positive integer, n is a positive integer , We fix a vector space Fd . Consider the following sets. q The set X of n-step flags V = (V ) in Fd such that V = 0, V V . • i 0≤i≤n q 0 i ⊆ i+1 The set Y of complete flags F = (F ) in Fd such that F F , F = i . • i 0≤i≤d q i ⊂ i+1 | i| where we write F for the dimension of F . i i Let G = GL(V| ) .| Then G acts naturallyon sets X andY . Moreoever, G acts transitively on Y . Let G act diagonally on the product X X (resp. X Y and Y Y ). Set × × × (6) = Z[v±1,t±1]. A Let (7) X = G(X X ) S A × be the set of all -valued G-invariant functions on X X . Clearly, the set X is a free A × S -module. Moreover, X admits an associative -algebra structure ‘ ’ under a standard A S A ∗ convolution product as discussed in [BKLW14, 2. 3]. In particular, when v is specialized to √q, we have (8) f g(V,V′) = f(V,V′′)g(V′′,V′), V,V′ X . ∗ ∀ ∈ V′′∈X X Similarly, we define the free -modules A (9) = G(X Y ) and Y = G(Y Y ). V A × H A × A similar convolution product gives an associative algebra structure on Y and a left X- H S action and a right Y-action on . Moreover, these two actions commute and hence we H V have the following -algebra homomorphisms. A SX → EndHY(V) and HY → EndSX(V). Similar to [P09, Theorem 2. 1], we have the following double centralizer property. Lemma 3.1.1. EndHY(V) ≃ SX and EndSX(V) ≃ HY, if n ≥ d. 6 HAITAOMA, ZONGZHULIN,ANDZHU-JUNZHENG We note that the result in [P09, Theorem 2. 1] is obtained over the field C of complex numbers, but the proof can be adapted to our setting over the ring . We shall give a description of the G-orbits on X X , X Y anAd Y Y . We start by × × × introducing the following notations associated to a matrix M = (m ) . ij 1≤i,j≤c c ro(M) = m , ij ! j=1 (10) X 1≤i≤c c co(M) = m . ij ! Xi=1 1≤j≤c We also write ro(M) and co(M) for the i-th and j-th component of the row vectors of i j ro(M) and co(M), respectively. For any pair (V,V′) of flags in X , we can assign an n by n matrix whose (i,j)-entry equal to dim Vi−1+Vi∩Vj′ . Vi−1+Vi∩Vj′−1 (11) G X X Θ , d \ × ≃ where Θ is the set of all matrices Θ in Mat (N) such that (Θ ) = d d d n×n i,j d i,j A similar assignment yields two bijection P (12) G X Y Π, \ × ≃ (13) G Y Y Σ, \ × ≃ where the set Π consists of all matrices B = (b ) in Mat (N) subject to ij n×d co(B) = 1, j [1,d]. j ∀ ∈ and Σ is the set of all matrices σ (σ ) in Mat (N) such that ij d×d ≡ ro(σ) = 1, ro(σ) = 1. i j Moreover, we have (14) #Σ = d! and #Π = nd. 4. Calculus of the algebra and Y S H Recall from the previous section that X is the convolution algebra on X X defined S × in (7). For simplicity, we shall denote instead of X. In this section, we determine the S S generators for and the associated multiplication formula. We also will Y action on . S H V GEOMETRIC SCHUR DUALITY OF TWO PARAMETER QUANTUM GROUP OF TYPE A 7 4.1. Defining relations of . For any i [1,n 1], a [1,n], set S ∈ − ∈ Ei(V,V′) = v−|Vi′/Vi′−1|t−|Vi/Vi−1|, if Vi ⊃1 Vi′,Vj = Vj′,∀j ∈ [1,n]\{i}; (0, otherwise. Fi(V,V′) = v−|Vi′+1/Vi′|t|Vi′+1/Vi′|, if Vi ⊂1 Vi′,Vj = Vj′,∀j ∈ [1,n]\{i}; (0, otherwise. (15) v±|Va′/Va′−1|t±|Va′/Va′−1|, if V = V′; A±1(V,V′) = a 0, otherwise. ( v∓|Va′/Va′−1|t±|Va′/Va′−1|, if V = V′; B±1(V,V′) = a 0, otherwise. ( It is clear that these functions are elements in . S Proposition 4.1.1. The functions E ,F , A±1 and B±1 in , for any i [1,n 1], a [1,n], i i a a S ∈ − ∈ satisfy the following relations. (R1) A±1A±1 = A±1A±1, B±1B±1 = B±1B±1, i j j i i j j i A±1B±1 = B±1A±1, A±1A∓1 = 1 = B±1B∓1. i j j i i i i i (R2) A E A−1 = vhi,jithi,jiE , B E B−1 = v−hi,jithi,jiE , i j i j i j i j A F A−1 = v−hi,jit−hj,iiF , B F B−1 = vhi,jit−hj,iiF . i j i j i j i j A B B A i i+1 i i+1 (R3) E F F E = δ − . i j − j i ij v v−1 − (R4) (−1)pt−p(p′−2hii,·jii+2hji·,iii)Ei(p′)EjEi(p) = 0, if i 6= j, p+p′X=1−2ii··ji (−1)pt−p(p′−2hii,·jii+2hji·,iii)Fi(p)FjFi(p′) = 0, if i 6= j, p+p′X=1−2ii··ji n n (R5) A = vdtd, B = v−dtd; i i i=1 i=1 Y Y d d (R6) (A vltl) = 0, (B v−ltl) = 0 j [1,n]. j j − − ∀ ∈ l=0 l=0 Y Y (R7) E d+1 = 0,F d+1 = 0. i i 8 HAITAOMA, ZONGZHULIN,ANDZHU-JUNZHENG Proof. The proofs of the identities of R1, R7 are straightforward. Let λ′ = V′/V′ . We i | i i−1| show the first identity in R2. we have vλ′i−λ′j−1tλ′i+λ′j if V 1 V′and i = j +1, j ⊃ j (A E )(V,V′) =  vλ′i−λ′j+1tλ′i+λ′j+2 if Vj ⊃1 Vj′and i = j, i j   vλ′i−λ′jtλ′i+λ′j+1 if V 1 V′and i = j,j +1, j ⊃ j 6 0 otherwise.       vλ′i−λ′jtλ′i+λ′j+1 if V 1 V′, (A E )(V,V′) = j ⊃ j i j ( 0 otherwise. That is, A E A−1(V,V′) = vhi,jithi,jiE (V,V′). All other identities can be shown similarly. i j i j we show the identity in R3. By a direct calculation. We have vλ′i−λ′i+1tλ′i+λ′i+1−vλ′i+1−λ′itλ′i+λ′i+1 if V = V′andi = j, (E F F E )(V,V′) = v−v−1 i j j i − ( 0 otherwise. It is easy to check that the right hand side is equal to δ AiBi+1−BiAi+1(V,V′). ij v−v−1 At last, We now show the first identity in R4. By a direct calculation, we have Ei2Ei+1(V,V′) = (v2 +1)v−2λ′i−λ′i+1−1t2λ′i+λ′i+1+4, if Vi ⊃2 Vi′and Vi+1 ⊃1 Vi′+1, ( 0, otherwise. EiEi+1Ei(V,V′) = (v2 +1)v−2λ′i−λ′i+1t2λ′i+λ′i+1+3, if Vi ⊃2 Vi′and Vi+1 ⊃1 Vi′+1, ( 0, otherwise. Ei+1Ei2(V,V′) = (v2 +1)v−2λ′i−λ′i+1+1t2λ′i+λ′i+1+2, if Vi ⊃2 Vi′and Vi+1 ⊃1 Vi′+1, ( 0, otherwise. The first identity in R4 follows. By the same way, the other three identities can be shown directly. Let’s prove the first identity in R5, we have n vλ′1+···+λ′ntλ′1+···+λ′n, if V = V′, A (V,V′) = i 0, otherwise. i=1 (cid:26) Y Since λ′ + + λ′ = d, the first identities follows. The other identities can be shown 1 ··· n similarly. At last, let’s prove the first identity in R6, we have d (Aj −vltl)(V,V′) = 0(v,λ′jtλ′j −1)(vλ′jtλ′j −vt)...(vλ′jtλ′j −vdtd), oifthVer=wiVse′., l=0 (cid:26) Y Since 0 λ′ d, the first identities follows. the other identities can be shown similarly. (cid:3) ≤ j ≤ GEOMETRIC SCHUR DUALITY OF TWO PARAMETER QUANTUM GROUP OF TYPE A 9 4.2. Multiplication formulas in . For any n Z,k N, set S ∈ ∈ v2n 1 n k (n+1 i) v (n) = − , and = − . v v2 1 k (i) − (cid:18) (cid:19)v i=1 v Y Let E is the n n matrix whose (i,j)-entry is 1 and all other entries are 0. Let e be ij a × the characteristic function of the G-orbit corresponding to a θ . It is clear that the set d ∈ e a θ forms a basis of . a d { | ∈ } S We assume that the ground field is an algebraic closure F of F when we talk about the q q dimension of a G-orbit or its stabilizer. Set d(a) = dim and r(a) = dim , a θ or Π, a b d O O ∀ ∈ where b = (b ) is the diagonal matrix such that b = a . Denote by C (V,V′) the ij ii k ik G stabilizer of (V,V′) in G. P Lemma 4.2.1. If a Π , We have ∈ dim C (V,V′) = a a , if (V,V′) , G ij kl a ∈ O i≥k,j≥l X dim = a a , a ij kl O il jl ju ul j

Geometric Schur Duality of two parameter quantum group of type A

Similar Geometric Schur Duality of two parameter quantum group of type A