Proceedings of Symposia in Pure Mathematics

Structures and Representations of Affine q-Schur Algebras

Jie Du

Abstract. This paper provides a survey for the latest developments in thetheory of affine q-Schur algebras and Schur–Weyl duality between affine quan-tum gln and affine type A Hecke algebras. More precisely, we will establish, onthe one side, an isomorphism between the double Ringel–Hall algebra D(n)of a cyclic quiver (n) and the quantum loop algebra of gln, and establish,on the other side, explicit epimorphisms from the double Ringel–Hall algebrato affine q-Schur algebras S(n, r). Then, by two commuting actions on anaffine tensor space, we will establish, for n > r, a category equivalence be-tween representations of affine q-Schur algebras and representations of affineHecke algebras. In this way, we describe two classification theorems for sim-ple representations of affine q-Schur algebras over C when the parameter q isnot a root of unity. Further structures of affine q-Schur algebras will also bediscussed as applications of the epimorphisms mentioned above.

1. Introduction

It is well known that Schur algebras or their quantum analogue—q-Schuralgebras—are a class of finite dimensional algebras which play an important role inthe theory of Schur–Weyl duality. This theory links representations of the generalor quantum linear group with those of the symmetric group or the associated Heckealgebra. Over its history of more than one hundred years, it continues to make pro-found influences in several areas of mathematics such as Lie theory, representationtheory, invariant theory, combinatorics, etc. Recent new developments include, e.g.,walled Brauer algebras and rational q-Schur algebras arising from quantum gln ac-tions on mixed tensor spaces [11], quantum Schur superalgebras [16], quiver Schuralgebras [37], and integral Schur–Weyl duality for type C [32], and so on. Thispaper provides a survey for the latest work [8] by Deng, Fu, and the author in thestudy of affine q-Schur algebras and Schur–Weyl duality between affine quantumgln and affine type A Hecke algebras.

The introduction of the affine analogue of q-Schur algebras was given byGinzburg–Vasserot [20] in 1993 as a generalization of the geometric approach toq-Schur algebras developed by Beilinson–Lusztig–MacPherson [2]. Later, in 1999,

2010 Mathematics Subject Classification. Primary 17B37, 20G43, 20C08; Secondary 16G20,16T20.

Supported partially by Australian Research Council.

c©0000 (copyright holder)

1

2 JIE DU

several other versions have appeared in [29], [38] (see also [22] for another defi-nition). Independently, representations of affine Hecke algebras H(r)C when theparameter q is not a root of unity have been investigated much earlier by Zelevinsky[42] in 1983 and Rogawski [35] in 1985. On the other hand, besides the standardrepresentation theory of quantum groups (associated with symmetrizable Cartanmatrices) as developed in [28], there is, in the affine type case, a so-called pseudo-highest weight module theory developed by Chari–Pressley around 1994 and itsgeneralization to affine quantum gln (or more precisely the quantum loop algebraof gln) by Frenkel–Mukhin [17]. Though some connections under the assumption

n > r (resp., n > r) between representations of UC(sln) (resp., UC(gln)) and H(r)Cwere observed in [6] (resp., [19]), it was unclear until the work [8] how the affine q-Schur algebra S(n, r)C plays a bridging role between affine quantum gln and affineHecke algebras:

UC(gln)-Mod H(r)C-Mod

S(n, r)C-Mod

In this paper, we will present this bridging relation through a discussion on thestructures and representations of affine q-Schur algebras.

To build this bridging relation, we need a UC(gln)-H(r)C-bimodule structureon the affine tensor space Ω⊗r

C . Using the geometric description of the Hall algebraH(n)C of a cyclic quiver (n), Varagnolo–Vasserot have already investigated an

H(n)opC -H(r)C-bimodule structure on Ω⊗r

C . Motivated by works of Schiffmann [36]and Hubery [24], we introduced the double Ringel–Hall algebra D,C(n) togetherwith a Drinfeld–Jimbo type presentation involving Chevalley and (infinitely many)central generators. Thus, the quantum affine sln is the subalgebra generated bythe Chevalley generators. By extending an isomorphism of Beck [1] between thequantum affine sln of Drinfeld–Jimbo type and the quantum loop algebra of sln withDrinfeld’s new presentation [13], we obtain a Hopf algebra isomorphism between

D,C(n) and UC(gln) (see Theorem 2.6). In this way, the natural action of D,C(n)

on the affine tensor space Ω⊗rC , which extends the action of Varagnolo–Vasserot,

gives rise to an action of UC(gln) on Ω⊗rC and, hence, a UC(gln)-H(r)C-bimodule

structure on Ω⊗rC .

The double Ringel–Hall algebra has naturally a triangular decomposition withthe ±-parts isomorphic to the Hall algebra. Thus, by an analysis of a spanning set ofBeilinson–Lusztig–MacPherson (BLM) type for affine q-Schur algebras, we managedto establish explicitly an epimorphism from the double Ringel–Hall algebra to affineq-Schur algebras. In this way, the structures of affine q-Schur algebras are nicelypresented through its triangular decomposition, integral bases, central subalgebras,and generators/relations (for n > r). These material will form the first part ofthe paper. We will also mention a few conjectures at the end of Part 1 regardingLusztig type forms, BLM type realization for the double Ringel–Hall algebra, andthe second centralizer property in the affine quantum Schur–Weyl reciprocity.

Part 2 of the paper discusses the bridging relation between the representations.When the parameter is not a complex root of 1, two classifications for irreduciblerepresentations of affine q-Schur algebras will be determined. As mentioned above,they arise from the corresponding classifications for affine Hecke algebras H(r)C in

AFFINE q-SCHUR ALGEBRAS 3

terms of multisegments and for quantum loop algebras UC(gln) in terms of dominantDrinfeld polynomials. Moreover, we will identify them by establishing a bijectionfrom multisegments of each length 6 n and total length r to n-tuples of dominantDrinfeld polynomials of total degree r.

It would be interesting to point out that research on further applications ofthe work is underway. For example, irreducible representations of affine q-Schuralgebras at a complex root of 1 have been classified by Fu.1 Also, Drinfeld polyno-mials associated with small representations L(λ)a of S(n, r)C (a ∈ C∗) have beencomputed in terms of row residue sequences of partition λ.2

Some Notations. Let Z = Z[υ, υ−1] be the ring of integral Laurent polynomials,let Q(υ) be the fraction field of Z, and let C∗ = C\0. For integers N, t witht > 0, let

[N

t

]=

∏

16i6t

υN−i+1 − υ−(N−i+1)

υi − υ−i∈ Z and

[N

0

]= 1.

If we put [m] = υm−υ−m

υ−υ−1 =[m1

]and [N ]! := [1][2] · · · [N ], then

[Nt

]= [N ]!

[t]![N−t]! for

all 1 6 t 6 N . We also write [m]q etc. to be the evaluation at q ∈ C.In Part 1, algebras are mostly defined over Z or Q(υ). In particular, if an alge-

bra is defined over Z such as H(n), H(r), S(n, r) etc., then we will use the sameletter in bold face to denote the corresponding algebra H(n), H(r), S(n, r) etc.obtained by based change to Q(υ). In Part 2, we will mainly consider representa-tions over C for algebras H(r)C, S(n, r)C etc. which are obtained by specializingυ to a non-root of unity q ∈ C∗.

⋆ ⋆ ⋆

Acknowledgement. The paper was based on the two lectures given by the authorat the Southeastern Lie Theory Workshop, Charlottesville, June 1–4, 2011. I wouldlike to thank the organizers for the opportunity of attending this special conferencewhich marks the retirement of Professor Leonard Scott.

Leonard Scott was my postdoctoral advisor from 1988 to 1990, a crucial periodin my life. Not only he taught me mathematics, but also his generosity, his greatsense of humor, and his optimistic attitude towards life influenced me so much forso many years!

Part 1: The Structures

Let = (n) denote the cyclic quiver

b b b b b

bn

1 2 3 n−2 n−1

with vertex set I = Z/nZ = 1, 2, . . . , n and arrow set i→ i+1 | i ∈ I. Let F bea field. In the category of finite-dimensional nilpotent representations of over F,the (one-dimensional) simple objects are denoted by Si, i ∈ I, and indecomposable

1Q. Fu, Affine quantum Schur algebras at roots of unity, preprint 2012.2J. Du and Q. Fu, Small representations for affine q-Schur algebras, preprint (2012).

4 JIE DU

objects by Si[l], for all i ∈ I and l > 1. We will use the following matrix notationto denote the isomorphism classes (or isoclasses) of nilpotent representations.

For i, j ∈ Z, let Ei,j = (ek,l)k,l∈Z be the Z × Z-matrix defined by ek,l =

δk,i+snδl,j+sn (k, l, s ∈ Z). Let Θ(n) (resp., Θ+ (n)) be the N-span of E

i,j |1 6 i 6

n, j ∈ Z (resp., Ei,j |1 6 i 6 n, j ∈ Z, i < j. Let also Θ−

(n) = tA | A ∈ Θ+ (n).

Thus, any A = (ai,j) ∈ Θ(n) satisfies ai,j = ai+n,j+n for all i, j ∈ Z and each rowhas a finite sum. Let, for r > 0

Θ(n, r) = A ∈ Θ(n) | σ(A) = r,

Zn = (λi)i∈Z ∈ ZZ | λi = λi+n, ∀i ∈ Z, and

Λ(n, r) = λ ∈ Zn | λi > 0 ∀i, σ(λ) = r,

where σ(A) =∑n

i=1

∑j∈Z ai,j and σ(λ) = λ1 + · · · + λn. Then, we may identify

Λ(n, r) with the set Λ(n, r) of compositions of r into n parts. Note that takingsum of each row or column defines functions

ro, co : Θ(n) → Zn .

which map Θ(n, r) onto Λ(n, r).For any A =

∑16i6n

i<jai,jE

i,j ∈ Θ+

(n), define

M(A) =MF(A) = ⊕ 16i6ni<j

ai,jSi[j − i].

Then [M(A)] | A ∈ Θ+ (n) is a complete set of isoclasses of nilpotent representa-

tions of .

2. Double Ringel–Hall algebras and quantum loop algebras

For F = Fq, let hCA,B be the number of submodules N of MFq

(C) such that

MFq(C)/N ∼=MFq

(A) and N ∼=MFq(B). By a result of Ringel, there exist polyno-

mials ϕCA,B(υ

2), called Hall polynomials, such that ϕCA,B(q) = hCA,B.

Let H(n) be the (generic) Ringel–Hall algebra [33] associated with the cyclicquiver (n), which is by definition the free Z-module with basis uA = u[M(A)] |

A ∈ Θ+ (n). The multiplication is given by, for A,B ∈ Θ+

(n),

uAuB = υ〈d(A),d(B)〉∑

C∈Θ+ (n)

ϕCA,BuC

where d(D) denotes the dimension vector of M(D) and

〈d(A),d(B)〉 = dimHom(M(A),M(B)) − dimExt1(M(A),M(B))

is the Euler form associated with the cyclic quiver (n). If we write d(A) = (ai)i∈I

and d(B) = (bi)i∈I , the dimension vectors of M(A) and M(B), respectively, then

〈d(A),d(B)〉 =∑

i∈I

aibi −∑

i∈I

aibi+1.

The Z-subalgebra C(n) of H(n) generated by u[mSi] (i ∈ I and m > 1) iscalled the composition algebra of (n) (see [34]).

Theorem 2.1. Let C(n) = C(n)⊗Q(υ) and H(n) = H(n)⊗Q(υ).

(1) ([34]) The Q(υ)-algebra C(n) is isomorphic to the ±-part of the quantum

enveloping algebra of sln.

AFFINE q-SCHUR ALGEBRAS 5

(2) ([36]) There are (algebraically independent) central elements cm,m > 1,in H(n) such that

H(n) = C(n)⊗Q(υ) Q(v)[c1, c2, . . .].

Remark 2.2. The Ringel–Hall algebra H(n) has a grading H(n) =⊕d∈NnH(n)d by dimension vectors d. Each component H(n)d can be definedgeometrically in terms of invariant functions on a graded space ⊕i∈IVi withd = (dimVi)i under the conjugate action of

∏i∈I GL(Vi). This gives rise to a

geometric definition of Hall algebras and, hence, results in many other applica-tions. For example, Lusztig used this to introduce the canonical bases for quantumenveloping algebras.

Let

H(n)>0

= H(n)⊗Q(υ) Q(υ)[K±11 , . . . ,K±n

n ] and

H(n)60

= Q(υ)[K±11 , . . . ,K±n

n ]⊗Q(υ) H(n).

Putting x = x ⊗ 1 and y = 1 ⊗ y for x ∈ H(n) and y ∈ Q(υ)[K±11 , . . . ,K±n

n ],

H(n)>0

and H(n)60

are Q(υ)-spaces with bases u+AKα | α ∈ ZI, A ∈ Θ+ (n)

and Kαu−A | α ∈ ZI, A ∈ Θ+

(n), respectively. By work of Green [21] and Xiao

[40], it is known that both H(n)>0

and H(n)60

are Hopf algebras. Thus, one canbuild a larger Hopf algebra by using a so-called skew-Hopf pairing [26].

By [40, Prop. 5.3], the Q(υ)-bilinear form ψ : H(n)>0 × H(n)

60 → Q(υ)defined by

ψ(u+AKα,Kβu−B) = υαβ−〈d(A),d(A)+α〉+2d(A)a−1

A δA,B,

where α, β ∈ ZI and A,B ∈ Θ+ (n), is a skew-Hopf pairing. Here, d(A) is the

dimension of M(A) and aA(q) = |AutMFq(A)|.

With ψ, we define I to be the ideal of the free product H(n)>0 ∗ H(n)

60

generated by∑

(b2 ∗ a2)ψ(a1, b1)−∑

(a1 ∗ b1)ψ(a2, b2) (a ∈ H(n)>0, b ∈ H(n)

60), 3

where ∆H(n)

>0(a) =∑a1 ⊗ a2 and ∆

H(n)60(b) =

∑b1 ⊗ b2. Define the Drinfeld

double by D(n) := H(n)>0

∗H(n)60/I and the double Ringle–Hall algebra by

D(n) = D(n)/J = H(n)+·D(n)

0 ·H(n)−,

where J denotes the ideal generated by 1⊗Kα −Kα ⊗ 1 for all α ∈ ZI, H(n)+=

H(n), H(n)−= H(n)

op, and D(n)

0 is the subalgebra generated by the K±i .

Remark 2.3. If q ∈ C∗ is not a root of unity, then aA(q) 6= 0. Thus, the Hopfpairing is well defined over C with respect to such q. So the double Ringel–Hallalgebra D,C(n) is defined over C.

The following result is built on the Hubery’s explicit description [23, 24] ofSchiffmann’s central elements ci. Let (ci,j)i,j be the Cartan matrix of affine typeA.

3In fact, it is enough to take a, b to be all semisimple generators u+a , u−

b, respectively; see [8,

2.6.1].

6 JIE DU

Theorem 2.4 ([8, 2.3.1,2.3.5]). The double Ringel–Hall algebra D(n) ofthe cyclic quiver (n) is the Hopf Q(υ)-algebra generated by Ei = u+[Si]

, Fi =

u−[Si], Ki, K

−1i , z

+s , z

−s , i ∈ I, s ∈ Z+ with relations (i, j ∈ I and s, t ∈ Z+):

(QGL1) KiKj = KjKi, KiK−1i = 1;

(QGL2) KiEj = vδi,j−δi,j+1EjKi, KiFj = v−δi,j+δi,j+1FjKi;

(QGL3) EiFj − FjEi = δi,jKi−K

−1i

υ−υ−1 , where Ki = KiK−1i+1;

(QGL4)∑

a+b=1−ci,j

(−1)a[1− ci,ja

]Ea

i EjEbi = 0 for i 6= j;

(QGL5)∑

a+b=1−ci,j

(−1)a[1− ci,ja

]F ai FjF

bi = 0 for i 6= j;

(QGL6) z+s z

+t = z

+t z

+s ,z

−s z

−t = z

−t z

−s , z

+s z

−t = z

−t z

+s ;

(QGL7) Kiz+s = z

+s Ki, Kiz

−s = z

−s Ki;

(QGL8) Eiz+s = z

+s Ei, Eiz

−s = z

−s Ei, Fiz

−s = z

−s Fi, and z

+s Fi = Fiz

+s .

Moreover, the Hopf algebra structure is given by comultiplication ∆, counit ε,and antipode σ defined by

∆(Ei) = Ei ⊗ Ki + 1⊗ Ei, ∆(Fi) = Fi ⊗ 1 + K−1i ⊗ Fi,

∆(K±1i ) = K±1

i ⊗K±1i , ε(Ei) = ε(Fi) = 0, ε(Ki) = 1;

σ(Ei) = −EiK−1i , σ(Fi) = −KiFi, σ(K±1

i ) = K∓1i ,

∆(z±s ) = z±s ⊗ 1 + 1⊗ z

±s ε(z±s ) = 0,

and σ(z±s ) = −z±s ,

where i ∈ I and s ∈ Z+.Replacing Q(υ) by C and υ by a non-root of unity q, a similar result holds for

D,C(n).

The subalgebra U(n) of D(n) generated by Ei, Fi,Ki,K−1i (i ∈ I) is the

(extended) quantum affine sln, which contains the subalgebra U(sln) generated by

Ei, Fi, Ki, K−1i (i ∈ I). The latter is the quantum enveloping algebra of the loop

algebra of sln.Motivated from the universal enveloping algebra of a loop algebra, Drinfeld

introduced the following quantum version in [13].

Definition 2.5. (1) The quantum loop algebra U(gln) (or quantum affine gln)is the Q(υ)-algebra generated by x

±i,s (1 6 i < n, s ∈ Z), k±1

i and gi,t (1 6 i 6 n,

t ∈ Z\0) with the following relations:

(QLA1) kik−1i = 1 = k

−1i ki, [ki, kj] = 0;

(QLA2) kix±j,s = v±(δi,j−δi,j+1)x

±j,ski, [ki, gj,s] = 0;

(QLA3) [gi,s, x±j,t] =

0 if i 6= j, j + 1,

±v−is [s]sx±i,s+t, if i = j,

∓v(1−i)s [s]sx±i−1,s+t, if i = j + 1;

(QLA4) [gi,s, gj,t] = 0;

(QLA5) [x+i,s, x−j,t] = δi,j

φ+i,s+t−φ

−

i,s+t

v−v−1 ;

(QLA6) x±i,sx

±j,t = x

±j,tx

±i,s, for |i−j| > 1, and [x±i,s+1, x

±j,t]v±cij = −[x±j,t+1, x

±i,s]v±cij ;

(QLA7) [x±i,s, [x±j,t, x

±i,p]v]v = −[x±i,p, [x

±j,t, x

±i,s]v]v for |i− j| = 1,

AFFINE q-SCHUR ALGEBRAS 7

where [x, y]a = xy − ayx, and φ±i,s are defined by the generating functions in inde-terminate u:

(2.1) Φ±i (u) := k

±1i exp(±(v − v−1)

∑

m>1

hi,±mu±m) =

∑

s>0

φ±i,±su±s.

Here ki = ki/ki+1 (kn+1 = k1) and hi,m = v(i−1)mgi,m − v(i+1)mgi+1,m, for all1 6 i < n.

(2) Let UC(gln) be the quantum loop algebra defined by the same generatorsand relations (QLA1)–(QLA7) with Q(υ) replaced by C and υ by a non-root ofunity q ∈ C.

Set, for each s > 1,

θ±s = ∓1

[s](g1,s + · · ·+ gn,s).

The following result is a lifting of Beck’s embedding [1] from quantum sln into

U(gln) and Hubery’s embedding [24] of the extended Ringel–Hall algebrasH(n)>0

into U(gln).

Theorem 2.6 ([8, 2.5.3]). There is a Hopf algebra isomorphism E : D(n) −→

U(gln) such that

K±1i 7−→ k

±1i , Ej 7−→ x

+j,0, Fj 7−→ x

−j,0 (1 6 i 6 n, 1 6 j < n),

En 7−→ ε+n , Fn 7−→ ε−n , z±s 7−→ θ±s (s > 1),

where

ε+n = (−1)n[x−n−1,0, · · · , [x−3,0, [x

−2,0, x

−1,1]υ−1 ]υ−1 · · · ]υ−1 kn and

ε−n = (−1)nk−1n [· · · [[x+1,−1, x

+2,0]υ, x

+3,0]υ, · · · , x

+n−1,0]υ.

Similarly, with υ replaced by a non-root of unity q ∈ C∗, there is a C-algebra

isomorphism EC : D,C(n) → UC(gln).

This isomorphism is a key ingredient for the affine q-Schur algebra S(n, r)C to

link representations of the quantum loop algebra UC(gln) with those of the affineHecke algebra H(r)C of type A. For this purpose, we adjust E to the Hopf algebra

automorphism βC : D,C(n) −→ UC(gln) satisfying

(2.2)K±1

i 7−→ k±1i , Ej 7−→ x

+j,0, Fj 7−→ x

−j,0 (1 6 i 6 n, 1 6 j < n),

En 7−→ (−1)nυε+n , Fn 7−→ (−1)nυ−1ε−n , z±s 7−→ ∓sυ±sθ±s (s > 1).

3. Affine q-Schur algebras

We now construct algebra epimorphisms from D(n) to affine q-Schur algebrasS(n, r) via the tensor spaces Ω⊗r.

Let Ω be the free Z-module with basis ωi | i ∈ Z and let Ω = Ω ⊗Z Q(υ).For i ∈ I and t ∈ Z+, we define the actions of Ei, Fi, K

±1i , z+t , and z

−t on Ω by

(3.1)Ei · ωs = δi+1,sωs−1, Fi · ωs = δi,sωs+1, K±1

i · ωs = υ±δi,sωs,

z+t · ωs = ωs−tn, and z

−t · ωs = ωs+tn.

8 JIE DU

It is direct to check that Ω becomes a left D(n)-module. Hence, the Hopf structureon D(n) induces a D(n)-module structure on Ω⊗r. Thus, we obtain an algebrahomomorphism

(3.2) ξr : D(n) → End(Ω⊗r).

Like the quantum gln case, the affine Hecke algebra H(r) over Q(υ) (see definitionbelow) acts on Ω⊗r and affine q-Schur algebras are the endomorphism algebras ofthis H(r)-module. We now describe this action.

Let S,r be the affine symmetric group consisting of all permutations w : Z → Z

such that w(i+ r) = w(i) + r for i ∈ Z. There are several useful subgroups of S,r.The subgroupW of S,r consisting of w ∈ S,r with

∑ri=1 w(i) =

∑ri=1 i is the

affine Weyl group of type A with generators si (1 6 i 6 r) defined by

si(j) =

j, for j 6≡ i, i+ 1 mod r;

j − 1, for j ≡ i+ 1 mod r;

j + 1, for j ≡ i mod r.

The cyclic subgroup 〈ρ〉 of S,r generated by the permutation ρ of Z sendingj to j + 1, for all j, is in the complement of W . Observe that sj+1ρ = ρsj , for allj ∈ Z and 1 6 j 6 n with sr+1 = s1.

The subgroup A of S,r consisting of permutations y of Z satisfying y(i) ≡i(mod r) is isomorphic to Zr via the map y 7→ λ = (λ1, λ2, . . . , λr) ∈ Zr, wherey(i) = λir + i, for all 1 6 i 6 r. Thus, A is identified with Zr. In particular, A isgenerated by ei = (0, . . . , 0, 1

(i), 0, . . . , 0) for 1 6 i 6 r.

Moreover, the subgroup of W generated by s1, . . . , sr−1 is isomorphic to thesymmetric group Sr. These groups are related by the following group isomor-phisms:

S,r∼= 〈ρ〉⋉W ∼= Sr ⋉ Zr.

The two isomorphisms give rise to two q-deformations of the group algebra ofS,r.

Following [25], the affine Hecke algebra H(r) := H(S,r) over Z is defined tobe the algebra generated by Tsi (1 6 i 6 r), T±1

ρ with the following relations:

T 2si

= (υ2 − 1)Tsi + υ2, TsiTsj = TsjTsi (i− j 6≡ ±1mod r),

TsiTsjTsi = TsjTsiTsj (i− j ≡ ±1mod r and r > 3),

TρT−1ρ = T−1

ρ Tρ = 1, and TρTsi = Tsi+1Tρ.

This algebra has a Z-basis Tww∈S,r, where Tw = T j

ρTsi1 · · ·Tsim if w =

ρjsi1 · · · sim is reduced. Let H(r) = H(r) ⊗Q(υ).Let H(r) = H(Sr) denote the Hecke algebra over Z associated with the sym-

metric group Sr. Then, H(r) is isomorphic to the subalgebra generated by Tsi(1 6 i 6 r − 1).

The Hecke algebraH(r) admits also the so-called Bernstein presentation whichconsists of generators

Ti := Tsi , Xj := Te1+···+ej−1 T−1e1+···+ej

(1 6 i 6 r − 1, 1 6 j 6 r),

AFFINE q-SCHUR ALGEBRAS 9

and relations

(Ti + 1)(Ti − υ2) = 0,

TiTi+1Ti = Ti+1TiTi+1, TiTj = TjTi (|i − j| > 1),

XiX−1i = 1 = X−1

i Xi, XiXj = XjXi,

TiXiTi = υ2Xi+1, and XjTi = TiXj (j 6= i, i+ 1).

Use the Bernstein presentation, Varagnolo–Vasserot defined an action of H(r)on the tensor space Ω⊗r in [38, 8.2(8)(9)] by

ωi ·X−1t = ωi1 · · ·ωit−1ωit+nωit+1 · · ·ωir , for all i ∈ Zr

ωi · Tk =

υ2ωi, if ik = ik+1;

vωisk , if ik < ik+1; for all i ∈ I(n, r),

vωisk + (υ2 − 1)ωi, if ik+1 < ik,

where 1 6 k 6 r − 1, 1 6 t 6 r, and I(n, r) = (i1, i2, . . . , ir) | 1 6 ij 6 n, ∀j. Let

S(n, r) := EndH(r)(Ω⊗r) and S(n, r) := EndH(r)(Ω

⊗r).

Both algebras S(n, r) and S(n, r) are called affine quantum Schur (or q-Schur)algebras over Z and Q(υ), respectively.

Proposition 3.1 ([8, (3.5.4)]). The action of D(n) on Ω⊗r commutes with theH(r) action of Varagnolo–Vasserot. Hence, (3.2) induces algebra homomorphisms

ξr : D(n) → EndH(r)(Ω⊗r) = S(n, r) and ξ∨r : H(r) → EndS(n,r)(Ω

⊗r)op.

Similar statements hold for D,C(n) and H(r)C.

Like the q-Schur algebra case, by specializing υ2 to a prime power q, the algebrasS(n, r)C and H(r)C and the tensor space Ω⊗r

C have geometric descriptions in termsof varieties of filtrations of lattices or cyclic flags (see [20], [29], and [38]), wheremultiplication is a convolution product. This multiplication induces an S(n, r)C-H(r)C-bimodule structure on Ω⊗r

C (see [29, 1.10], [38, 7.4]).

Remark 3.2. Using the geometric definition of Hall algebras mentioned inRemark 2.2, Varagnolo–Vasserot introduced in [38, Prop. 7.6] an algebra homo-

morphism from H(n)opC to S(n, r)C. This induces a (geometric) action of H(n)

−C

on the tensor space Ω⊗rC . It is proved in [8, §3.6] that this action coincides with

restriction of the action defined by (3.1).

Similar to q-Schur algebras, there is a standard (or defining) basis eAA∈Θ(n,r)

(notation of [29, 1.9]). For A ∈ Θ(n, r), let

(3.3) [A] = υ−dAeA, where dA =∑

16i6ni>k,j<l

ai,jak,l.

(See [29, 4.1(b),4.3] for a geometric meaning of dA.)

4. The Beilinson–Lusztig–MacPherson type construction

Let

Θ± (n) := A ∈ Θ(n) | ai,i = 0 for all i.

10 JIE DU

Following [2], for A ∈ Θ± (n) and j ∈ Zn

, define A(j, r) ∈ S(n, r) by

A(j, r) =

∑λ∈Λ(n,r−σ(A)) υ

λj[A+ diag(λ)], if σ(A) 6 r;

0, otherwise,

where λ · j =∑

16i6n λiji.

The set A(j, r) | A ∈ Θ± (n), j ∈ Zn

forms a spanning set for S(n, r). Explicitbases can be extracted; see [14, Prop. 4.1] or [8, (3.4.1)].

The following four theorems in this and next sections summarize some struc-tures of affine q-Schur algebras.

Theorem 4.1. (1)([8, Prop. 3.7.1]) There is a partial ordering 4 on Θ± (n)

such that, for any A = A+ +A− ∈ Θ± (n) with A+ ∈ Θ+

(n) and A− ∈ Θ− (n),

A+(0, r)A−(0, r) = A(0, r) +∑

B∈Θ± (n)

B≺A,j∈Zn

gB,j,A;rB(j, r) (in S(n, r)).

(2)([14, Th. 4.2]) There are explicit multiplication formulas for the products

O(j, r)A(j′, r), Eh,h+1(0, r)A(j, r), and E

h+1,h(0, r)A(j, r),

where O stands for the zero matrix. More precisely, for i ∈ Z, j, j′ ∈ Zn and

A = (ai,j) ∈ Θ± (n), if we put f(i) = f(i, A) =

∑j>i ah,j −

∑j>i ah+1,j and

f ′(i) = f ′(i, A) =∑

j<i ah,j −∑

j6i ah+1,j, then the following identities hold in

S(n, r) for all 1 6 h 6 n, r > 0:

O(j, r)A(j′, r) = υjro(A)A(j+ j′, r), A(j′, r)O(j, r) = υjco(A)A(j+ j′, r),

Eh,h+1(0, r)A(j, r) =

∑

i<h;ah+1,i>1

υf(i)[[ah,i + 1

1

]](A+ E

h,i − Eh+1,i)(j+α

h , r)

+∑

i>h+1;ah+1,i>1

υf(i)[[ah,i + 1

1

]](A+ E

h,i − Eh+1,i)(j, r)

+ υf(h)−jh−1(A− E

h+1,h)(j+αh , r) − (A− E

h+1,h)(j+ βh , r)

1− υ−2

+ υf(h+1)+jh+1

[[ah,h+1 + 1

1

]](A+ E

h,h+1)(j, r),

and

Eh+1,h(0, r)A(j, r) =

∑

i<h;ah,i>1

υf′(i)

[[ah+1,i + 1

1

]](A− E

h,i + Eh+1,i)(j, r)

+∑

i>h+1;ah,i>1

υf′(i)

[[ah+1,i + 1

1

]](A− E

h,i + Eh+1,i)(j −α

h , r)

+ υf′(h+1)−jh+1−1

(A− Eh,h+1)(j −α

h , r)− (A− Eh,h+1)(j+ β

h , r)

1− υ−2

+ υf′(h)+jh

[[ah+1,h + 1

1

]](A+ E

h+1,h)(j, r).

AFFINE q-SCHUR ALGEBRAS 11

(By convention, if an off-diagonal entry of A is negative, then A(j) = 0.)By the triangular relation in Theorem 4.1(1), we can establish a “triangular

decomposition” for S(n, r), which is the key to proving the surjectivity of the mapξr in Proposition 3.1.

Theorem 4.2 ([8, 3.7.7,3.8.1]). Let

S(n, r)± = spanZA(0, r) | A ∈ Θ±

(n) and

S(n, r)0 = spanZ[diag(λ)] | λ ∈ Λ(n, r).

Then S(n, r) = S(n, r)+S(n, r)

0S(n, r)− and the set

A+(0, r)[diag(λ(A))]A−(0, r) | A ∈ Θ(n, r),

where λ(A) = (ai,i +∑

j<i(ai,j + aj,i))i ∈ Λ(n, r) and A± is the upper/lower trian-

gular part of A, forms a Z-basis for S(n, r). Moreover, the algebra homomorphisms

ξr : D(n) → S(n, r) and ξr,C : D,C(n) → S(n, r)C

are surjective and, when n > r and F is a field which is a Z-algebra, the homomor-phism

ξ∨r,F : H(r)F → EndS(n,r)F(Ω⊗rF )op.

is an isomorphism.

Remark 4.3. The affine q-Schur algebra S(n, r)C can also be defined as thecomplexified Grothendieck group KG(Z) of G-equivariant coherent sheaves on the

Steinberg variety Z, see [20, (9.4)]. In [39, 2.4], an epimorphism from UC(gln) toKG(Z) is geometrically constructed.

5. Presenting affine q-Schur algebras S(n, r) for n > r

Let

U(n, r) := ξr(U(n)) and Z(n, r) := ξr(Z(n)),

where Z(n) = Q(υ)[z+m, z−m]m>1 is a central subalgebra of D(n). The second part

of the following result is a q-analogue of a result of Yang in [41], while the thirdpart is built on a result of McGerty [30, 6.4,6.6] (cf. [12]).

Theorem 5.1. (1) Z(n, r) is a central subalgebra of S(n, r) and S(n, r) =U(n, r)Z(n, r).

(2) Z(n, r) ∼= Q(υ)[σ1, . . . , σr−1, σr, σ−1r ] is a Laurent polynomial ring, and

Z(n, r) ⊂ U(n, r) if and only if n > r.(3)([8, Th. 5.1.1]) The algebra U(n, r) can be presented by generators

ei, fi, ki (i ∈ I) and relations: for all i, j ∈ I,

(QS1) kikj = kjki;(QS2) kiej = υδi,j−δi,j+1ejki, kifj = υ−δi,j+δi,j+1fjki;

(QS3) eifj − fjei = δi,jki−k

−1

i

υ−υ−1 , where ki = kik−1i+1;

(QS4)∑

a+b=1−ci,j

(−1)a[1− ci,ja

]eai eje

bi = 0 for i 6= j;

(QS5)∑

a+b=1−ci,j

(−1)a[1− ci,ja

]fai fjf

bi = 0 for i 6= j;

(QS6) [ki; r + 1]! = 0, k1 · · · kn = υr.

12 JIE DU

It seems to us that finding presentations for S(n, r) and n 6 r is too compli-cated. We couldn’t get the general case except the following n = r case.

Let ρ be the Q(υ)-linear isomorphism

ρ : Ω⊗r −→ Ω⊗r, ρ(ωi1 ⊗ · · · ⊗ ωir ) = ωi1−1 ⊗ · · · ⊗ ωir−1.

Then, ρ = ξr(∑

a∈Nr,σ(a)=r u+a ) = e′δ + eδ (i.e., the image of the sum of semisim-

ple modules of dimension r) and ρ−1 = ξr(∑

a∈Nr u−a ) = f′δ + fδ, where u±a =

υ∑

i ai(ai−1)u±a , eδ = ξr(u+δ ), and fδ = ξr(u

−δ ) with δ = (1, 1, . . . , 1).

Theorem 5.2. The Q(υ)-algebra S(r, r) is generated by ei, fi, k±1i (1 6 i 6 r)

and ρ±1 subject to the relations (QS1)–(QS6) above together with (QS0′)–(QS6′)below:

(QS0′) ρρ−1 = ρ−1ρ = 1, ρeiρ−1 = ei−1, ρfiρ

−1 = fi−1, ρkiρ−1 = ki−1;

(QS1′) (ki − υ)ρ = (ki − υ)e′δ;(QS2′) eiρ = eie

′δ +

1υ+υ−1 e

2i ei−1 · · · e1er · · · ei+1;

(QS3′) fiρ = fie′δ +

11−υ−2 ei−1 · · · e1er · · · ei+1k

−1i (ki+1 − k−1

i+1);

(QS4′) (ki − υ)ρ−1 = (ki − v)f′δ;(QS5′) fiρ

−1 = fif′δ +

1υ+υ−1 f

2i fi+1 · · · frf1 · · · fi−1;

(QS6′) eiρ−1 = eif

′δ +

11−υ−2 fi+1 · · · frf1 · · · fi−1k

−1i+1(ki − k−1

i ),

where (with the notation of quantum divided powers)

e′δ =

n∑

i=1

∑

a∈Nn

σ(a)=r,ai=0

(ei−1)(ai−1) · · · (e1)

(a1)(en)(an) · · · (ei+1)

(ai+1)

f′δ =

n∑

i=1

∑

a∈Nn

σ(a)=r,ai=0

(fi−1)(ai−1) · · · (f1)

(a1)(fn)(an) · · · (fi+1)

(ai+1).

In particular,

S(r, r) = U(r, r) ⊕⊕

06=m∈Z

Q(υ)ρm.

Note that (QS1′)–(QS6′) may be derived from the multiplication formulas givenin Theorem 4.1(2).

6. Some conjectures

We end this part with three conjectures. The first conjecture is a candidate of

a Lusztig type integral Z-form for the quantum loop algebra U(gln).

With the isomorphism between D(n) and U(gln), we introduce the Z-

subalgebra D(n)L of D(n) generated by K±1

i ,[Ki;0m

], u

±(m)i , and u±a for all i ∈ I,

m > 0, and sincere a = (ai) ∈ Nn (so all ai 6= 0). Here we used Hall algebranotation with u+i = Ei, u

−i = Fi and

[Ki; 0

m

]=

m∏

s=1

Kiυ−s+1 −K−1

i υs−1

υs − υ−sand

[Ki; 0

0

]= 1.

By [10, Th. 5.7], H(n)±is generated by u

±(m)i and u±a . This subalgebra is clearly

a Hopf subalgebra by [8, Prop. 3.5.6].

AFFINE q-SCHUR ALGEBRAS 13

If D(n)0 denotes the Z-subalgebra of D(n) generated by K±1

i and[Ki;0t

]for

i ∈ I and t > 0, then both H(n)+D(n)

0 and D(n)0H(n)

−are Hopf subalgebras.

Conjecture 6.1. We conjecture that

D(n)L = D(n) := H(n)

+D(n)

0H(n)− ∼= H(n)

+⊗D(n)

0 ⊗ H(n)−.

This is equivalent to saying that D(n) is a subalgebra. Note that the restriction

of ξr to D(n) is a surjective map onto S(n, r).

There is a smaller integral form D(n) generated by K±1i ,

[Ki;0m

], u

±(m)i , and

z±m. This form is contained in D(n) ([8, 3.7.5]) and does not map onto S(n, r) ([8,5.3.8]), but is useful for the reduction modulo υ = 1; see Chapter 6 in [8] for moredetails of this application.

In [17, 7.2], a so-called restricted integral form U resZ′ (gln) over Z ′ = C[υ, υ−1]

is defined as the Z ′-subalgebra generated by k±1i ,

[ki;0m

], x

±(m)i,s , and gi,t/[t], for all

i ∈ I, m > 0, s ∈ Z, and t ∈ Z\0. However, it is not clear if U resZ′ (gln) is a Hopf

subalgebra.The second conjecture is natural from the following observation. When r

approaches to ∞, the algebra epimorphism ξr : D(n) → S(n, r) implies that“limr→∞ S(n, r) = D(n)”. Thus, quantum affine gln can be realized as a limit ofaffine q-Schur algebras. In the non-affine case, this was established by Beilinson–Lusztig–MacPherson [2]. In fact, they discovered a stabilization property and con-

structed an algebra K (∼= U, the modified quantum gln), ‘larger’ than the directsum of all q-Schur algebras, whose completion (or limit) contains U(gln) as a sub-algebra. Thus, a new basis (called the BLM basis) for U(gln) was introducedand multiplication formulas between generators and arbitrary basis element weregiven. However, in the affine case, there is no stabilization property. So a modifiedapproach was developed in [14] which we now describe below.

For A ∈ Θ± (n) and j ∈ Zn

, let

A(j) = limr→∞

A(j, r) :=∑

λ∈Nn

υλj[A+ diag(λ)].

Clearly, A(j) =∑

r>0A(j, r) ∈ K(n) :=∏

r>0S(n, r).

Let A(n) be the subspace of K(n) spanned by A(j) for all A ∈ Θ± (n) and

j ∈ Zn . Since the structure constants (with respect to the BLM basis) appeared in

the multiplication formulas given in Theorem 4.1(2) are independent of r, takinglimits of both sides gives similar formulas in A(n). Thus, one sees easily that

U(n), H(n)>0

, H(n)60

as subalgebras are all in A(n).

Conjecture 6.2 ([14, 5.2(2)]). The Q(υ)-space A(n) is a subalgebra of K(n)

which is isomorphic to D(n) or the quantum loop algebra U(gln).

The conjecture has been established in the classical (υ = 1) case. In this case,explicit multiplication formulas for E

h,h+mn[0]A[j] (m ∈ Z\0), where Eh,h+mn[0]

are generators corresponding to homogeneous indecomposable modules, have beenfound; see [8, Ch. 6] for details (including notations).

The last conjecture is natural from the Schur–Weyl duality.

14 JIE DU

Conjecture 6.3. For n < r, the algebra homomorphisms

ξ∨r : H(r) −→ EndS(n,r)(Ω⊗r)op, ξ∨r,F : H(r)F −→ EndS(n,r)F(Ω

⊗rF )op

are surjective, where base change to an (infinite) field F is obtained by specializingυ to q ∈ F∗.

If q is a prime power, then Pouchin [31, Th. 8.1] has proved that ξ∨r,C is sur-

jective. Moreover, as proved in the non-affine case [15], it would be interestingto know if the kernel of ξ∨r,F can be described in terms of Kazhdan–Lusztig basiselements.

Part 2: The Representations

As before, we assume that q ∈ C∗ is not a root of unity. Since S(n, r) andH(r) are defined over Z, we may define S(n, r)F and H(r)F by base change to anyfield or ring F. By presentations similar to the case over Q(υ), we may define the

double Ringel–Hall algebras D,C(n) and the quantum loop algebras UC(gln) overC with respect to the parameter q; see 2.4 and 2.5. In this part, we investigate the

relationship between representations of UC(gln), S(n, r)C, and H(r)C.

7. A category equivalence

By Theorem 4.2, we have algebra epimorphism

ξr,C : D,C(n) −→ S(n, r)C

and algebra isomorphism (2.2)

βC : D,C(n) −→ UC(gln).

Hence, we obtain an algebra epimorphism

(7.1) ξ′r,C = ξr,C β−1C : UC(gln) −→ S(n, r)C.

Like the non-affine case, if n > r, then there is an idempotent eω ∈ S(n, r)Fsuch that H(r)F

∼= eωS(n, r)Feω, where F is a commutative ring which is a Z-algebra.

The first part of the following result together with the epimorphism (7.1) showsthat an affine q-Schur algebra links representations of the quantum loop algebra

UC(gln) with those of the Hecke algebra H(r)C.For an algebra A, let A-mod denote that category of finite dimensional left

A-modules.

Theorem 7.1. Let F be a field and let q ∈ F∗ be a non-root of unity.(1) ([8, Th 4.1.3]) If n > r, then there is a category equivalence

F : H(r)F-mod −→ S(n, r)F-mod,M 7−→ Ω⊗rF ⊗H(r)C N.

The inverse functor is the Schur functor G(M) = eωM for all M ∈ S(n, r)F-mod.(2) ([8, Th 4.1.6]) Every simple S(n, r)F-module is finite dimensional.

Note that part (2) is obtained by applying the first part to the fact that everysimple H(r)F-module is finite dimensional.

By this theorem, we will determine in §8 and §11 simple S(n, r)C-modulesthrough the classification of simple H(r)C-modules by Zelevinsky [42] and Ro-

gawski [35], as well as the classification of simple polynomial UC(gln)-modules byFrenkel–Mukhin [17], which is built on Chari–Pressley [3, 5].

AFFINE q-SCHUR ALGEBRAS 15

Remark 7.2. For F = C, a direct category equivalence from H(r)C-mod to a

full subcategory of UC(gln)-mod has been geometrically constructed in [19, Th. 6.8]when n > r. The simple objects in the full subcategory are described in terms ofintersection cohomology complexes.

8. Simple S(n, r)C-modules arising from simple H(r)C-modules

We first recall a classification of simple H(r)C-modules. Again, fix q ∈ C∗ suchthat qm 6= 1 for all m > 1.

A segment s with center a ∈ C∗ and length k is by definition an orderedsequence

s = (aq−k+1, aq−k+3, . . . , aqk−1) ∈ (C∗)k.

Write |s| = k and consider the set of all multisegments of total length r:

Sr = s = s1, . . . , sp | r = Σi|si| =: |s|.4

For s = s1, . . . , sp ∈ Sr, let

(a1, a2, . . . , ar) = (s1, . . . , sp) ∈ (C∗)r

be the r-tuple obtained by juxtaposing the segments in s and let Js be the left idealofH(r)C generated byXj−aj for 1 6 j 6 r. ThenMs = H(r)C/Js is a leftH(r)C-module which as an H(r)C-module is isomorphic to the regular representation ofH(r)C.

Let λ = (|s1|, . . . , |sp|). Then, the element

yλ =∑

w∈Sλ

(−q2)−ℓ(w)Tw ∈ H(r)C

generates the submodule H(r)Cyλ ofMs which, as an H(r)C-module, is isomorphicto H(r)Cyλ. Observe that

(8.1) H(r)Cyλ ∼= Eλ ⊕ (⊕

µ⊢r,µ⊲λ

mµ,λEµ),

where Eν is the left cell module defined by the Kazhdan–Lusztig’s C-basis [27]associated with the left cell containing w0,ν , the longest element of the Youngsubgroup Sν .

Let Vs be the unique composition factor of the H(r)C-module H(r)Cyλ suchthat the multiplicity of Eλ in Vs as an H(r)C-module is nonzero.

We now can state the following classification theorem due to Zelevinsky [42]and Rogawski [35]. The construction above follows [35].

Theorem 8.1. Let Irr(H(r)C) be the set of isoclasses of all simple H(r)C-modules. Then the correspondence s 7→ [Vs] defines a bijection from Sr toIrr(H(r)C).

Let

S(n)r = s ∈ Sr | Ω⊗r

C ⊗H(r)C Vs 6= 0.

Then S(n)r = Sr for all r 6 n. We have the following classification theorem; see

[8, 4.3.4&4.5.3].

4Here, possibly, si = sj for some i 6= j.

16 JIE DU

Theorem 8.2. The set Ω⊗rC ⊗H(r)C Vs | s ∈ S

(n)r is a complete set of

nonisomorphic simple S(n, r)C-modules. Moreover, we have5

(8.2) S(n)r = s = s1, . . . , sp ∈ Sr | p > 1, |si| 6 n ∀i.

In particular, if n > r, then the set Ω⊗rC ⊗H(r)C Vs | s ∈ Sr is a complete set of

nonisomorphic simple S(n, r)C-modules.

Thus, by ξ′r,C in (7.1), every Ω⊗rC ⊗H(r)C Vs is a simple UC(gln)-module. These

simple representations reflect a certain structure of the quantum loop algebra

UC(gln) hidden in the theory of so-called pseudo-highest weight representations[4, §12.2B].

9. Pseudo-highest weight representations

Recall Definition 2.5 and the generating function (2.1). For 1 6 j 6 n− 1 and

s ∈ Z, define the elements Pj,s ∈ UC(gln) through the generating functions

P±j (u) := exp

(−∑

t>1

1

[t]qhj,±t(qu)

±t

)=

∑

s>0

Pj,±su±s ∈ UC(gln)[[u, u

−1]].

A comparison with (2.1) gives

Φ±j (u)|υ=q = k

±1j

P±j (q−2u)

P±j (u)

.

For 1 6 i 6 n and s ∈ Z, define the elements Qi,s ∈ UC(gln) through thegenerating functions

Q±i (u) := exp

(−∑

t>1

1

[t]qgi,±t(qu)

±t

)=

∑

s>0

Qi,±su±s ∈ UC(gln)[[u, u

−1]].

Note that Q±i (u) and P

±j (u) are related by

P±j (u) =

Q±j (uq

j−1)

Q±j+1(uq

j+1), for all 1 6 j 6 n− 1.

Let V be a weight UC(sln)-module (resp., weight UC(gln)-module) of type 1.Then V = ⊕µ∈Zn−1Vµ (resp., V = ⊕λ∈ZnVλ) where

Vµ = v ∈ V | kjw = qµjw, ∀1 6 j < n

(resp., Vλ = v ∈ V | kjw = qλjw, ∀1 6 j 6 n).Following [4, 12.2.4], a nonzero weight vector w ∈ V is called a pseudo-highest

weight vector, if there exist some Pj,s ∈ C (resp. Qi,s ∈ C) such that

x+j,sw = 0, Pj,sw = Pj,sw, (resp., x+j,sw = 0, Qi,sw = Qi,sw).

for all s ∈ Z. The module V is called a pseudo-highest weight module if V =

UC(sln)w (resp., V = UC(gln)w) for some pseudo-highest weight vector w.

5The proof of this result requires Theorem 10.1 below.

AFFINE q-SCHUR ALGEBRAS 17

Remark 9.1. There are integrable highest weight modules considered in [28,6.2.3] and defined in [28, 3.5.6] which are different from pseudo-highest weightmodules defined here. This is seen as follows.

Let U±C be the subalgebra of UC(gln) generated by x±i,s for all i, s, and let U0

C

be the subalgebra of UC(gln) generated by all k±1i and Qi,s. Then UC(gln) =

U−CU

0CU

+C∼= U−

C ⊗ U0C ⊗ U+

C by [17, Lem 7.4]. If we identify the new realization of

UC(gln) with the double Hall algebra D,C(n) under the isomorphism βC given in(2.2), then En = ε+n ∈ U−

CU0C and Fn = ε−n ∈ U0

CU+C . If w0 is pseudo-highest weight

vector, then Uw0 = U−Cw0. Now Enw0 is not necessarily zero as En ∈ U−

C kn (whileFnw0 = 0). However, a highest weight vector w0 always satisfies Enw0 = 0.

Since all Qi,s commute with the kj , each Vλ is a direct sum of generalizedeigenspaces of the form

Vλ,γ = x ∈ Vλ | (Qi,s − γi,s)px = 0 for some p (1 6 i 6 n, s ∈ Z),

where γ = (γi,s) with γi,s ∈ C. Let Γ±j (u) =

∑s>0 γj,±su

±s.

A finite dimensional weight UC(gln)-module V (of type 1) is called a polynomialrepresentation if, for every weight λ = (λ1, . . . , λn) ∈ Zn of V , the formal powerseries Γ±

i (u) are polynomials in u± of degree λi so that the zeroes of the functionsΓ+i (u) and Γ−

i (u) are the same.Following [17], an n-tuple of polynomials Q = (Q1(u), . . . , Qn(u)) with

constant terms 1 is called dominant if, for 1 6 i 6 n − 1, the ratiosQi(q

i−1u)/Qi+1(qi+1u) are polynomials. Let Q(n) be the set of dominant n-tuples

of polynomials.For Q = (Q1(u), . . . , Qn(u)) ∈ Q(n), define Qi,s ∈ C (1 6 i 6 n, s ∈ Z) by the

following formula

Q±i (u) =

∑

s>0

Qi,±su±s.

Here f+(u) =∏

16i6m(1− aiu) ⇐⇒ f−(u) =∏

16i6m(1 − a−1i u−1).

Let I(Q) be the left ideal of UC(gln) generated by x+j,s,Qi,s−Qi,s, and ki−q

λi ,

for all 1 6 j 6 n− 1, 1 6 i 6 n and s ∈ Z, where λi = degQi(u), and define Vermatype module

M(Q) = UC(gln)/I(Q).

Then M(Q) has a unique simple quotient, denoted by L(Q). The polynomialsQi(u) are called Drinfeld polynomials associated with L(Q).

Similarly, for an (n−1)-tuples P = (P1(u), . . . , Pn−1(u)) ∈ P(n) of polynomialswith constant terms 1, define Pj,s ∈ C (1 6 j 6 n − 1, s ∈ Z) as in P±

j (u) =∑

s>0 Pj,±su±s and let µj = degPj(u). Replacing UC(gln), Qi,s − Qi,s, and ki −

qλi by UC(sln), Pi,s − Pi,s, and ki − qµi , respectively, in the above construction

defines a simple UC(sln)-module L(P). The polynomials Pi(u) are called Drinfeldpolynomials associated with L(P).

Theorem 9.2. (1)([3, 4]) Let P(n) be the set of (n− 1)-tuples of polynomialswith constant terms 1. The modules L(P) with P ∈ P(n) are all nonisomorphic

finite dimensional simple UC(sln)-modules of type 1.

18 JIE DU

(2)([17]) The UC(gln)-modules L(Q) with Q ∈ Q(n) are all nonisomorphic

finite dimensional simple polynomial representations of UC(gln). Moreover,

L(Q)|UC(sln)∼= L(P)

where P = (P1(u), . . . , Pn−1(u)) with Pi(u) = Qi(qi−1u)/Qi+1(q

i+1u).

We now use affine q-Schur algebras S(n, r)C to link these simple pseudo-

highest weight UC(gln)-modules with the simple UC(gln)-modules arising from sim-ple H(r)C-modules as given in Theorem 8.2.

10. Some Identification Theorems

By (7.1), every simple S(n, r)C-module of form Ω⊗rC ⊗H(r)CVs given in Theorem

8.2 is a simple UC(gln)-module. We now identify them as simple pseudo-highest

weight UC(gln)-modules under the assumption n > r and compute the associatedDrinfeld polynomials Qs.

Recall that Q(n) is the set of dominant n-tuples of polynomials. For r > 1, let

Q(n)r =Q = (Q1(u), . . . , Qn(u)) ∈ Q(n) | r =

∑

16i6n

degQi(u).

Let n > r. For s = s1, . . . , sp ∈ Sr, write

si = (aiq−νi+1, aiq

−νi+3, . . . , aiqνi−1) ∈ (C∗)νi ,

and define

Qs = (Q1(u), . . . , Qn(u))

by setting Qn(u) = 1 and, for 1 6 i 6 n− 1,

Q±i (u) = P±

i (uq−i+1)P±i+1(uq

−i+2) · · ·P±n−1(uq

n−2i),

where

P±i (u) =

∏

16j6pνj=i

(1− a±1j u±1).

If µi := degPi(u) = #j ∈ [1, p] | νj = i and

λi := degQi(u) = #j ∈ [1, p] | νj > i,

then λ = (λ1, . . . , λn−1, λn) is a partition of r and dual to (ν1, . . . , νp), and λi −λi+1 = µi for all 1 6 i < n.

Note that, if s ∈ S(n′)r (see (8.2)), where n′ < n, then Pn′+1(u) = · · · =

Pn−1(u) = 1. Thus, Qn′+1(u) = · · · = Qn(u) = 1. Thus, if n 6 r, we also define

(10.1) Qs = (Q1(u), . . . , Qn(u))

by dropping the 1’s.We have the following identification theorem.

Theorem 10.1 ([8, 4.4.2]). Maintain the notation above and let n > r.

(1) The map s 7→ Qs defines a bijection from Sr to Q(n)r.

(2) There are UC(gln)-module isomorphisms Ω⊗rC ⊗H(r)C Vs

∼= L(Qs) for alls ∈ Sr. Hence, the set L(Q) | Q ∈ Q(n)r forms a complete set ofnonisomorphic simple S(n, r)C-modules.

AFFINE q-SCHUR ALGEBRAS 19

The condition n > r simply means S(n, r)C is a homomorphic image of quan-tum affine sln. Thus, we may directly use the result [6, 7.6] together with a com-putation of the action of central elements to get the result. When n 6 r, differentargument is required; see remarks after Theorem 11.4. We first look at two exam-ples which will be used to establish this result in next section.

Example 10.2. Let a ∈ C∗.(1) Assume r = 1 < n. Then Ma = H(1)C/H(1)C(X1− a) is a simple H(1)C-

module and ΩC(a) := ΩC⊗Ma is a simple UC(gln)-module. IfQ = (1−au, 1, . . . , 1),then the theorem above implies that ΩC(a) ∼= L(Q).

(2) For 1 6 i 6 n− 1, define Qi,a ∈ Q(n) by setting Qn(u) = 1 and

Qj(uqj−1)

Qj+1(uqj+1)= (1 − au)δi,j

for 1 6 j 6 n− 1. In other words,

Qi,a = (1− aqi−1u, . . . , 1− aq−i+3u, 1− aq−i+1u(ith component)

, 1, . . . , 1).

Since n > i, by Theorem 10.1, the simple UC(gln)-module Li,a := L(Qi,a) is also asimple S(n, i)C-module. The weight of the pseudo-highest weight vector of Li,a isλ(i, a) = (1i, 0n−i).

We end this section with another identification for the 1-dimensional determi-nant representation of the affine q-Schur algebra S(n, n)C.

For any a ∈ C∗, let s = (a, aq2, . . . , aq2(n−1)) which is regarded as a singlesegment, and let

Deta := Ω⊗nC ⊗H(n)C Vs.

Then, Deta is the submodule of Ms generated by y(n) and is a simple S(n, n)C-module.

Proposition 10.3 ([8, 4.6.5]). The simple S(n, n)C-module Deta has dimen-sion 1 and

Deta = spanCω1 ⊗ · · · ⊗ ωn ⊗ y(n).

Moreover, as a UC(gln)-module, Deta ∼= L(Q), where Q = (Q1(u), . . . , Qn(u)) ∈Q(n) with Qi(u) = 1− aq2(n−i)u for all i = 1, 2, . . . , n.

11. A second classification of simple S(n, r)C-modules

By Example 10.2(1) together with the fact that Vs is a homomorphic image ofsome Ms, one can easily prove the following.

Lemma 11.1 ([8, 4.6.2]). Let V be a finite dimensional simple UC(gln)-module.Then the following conditions are equivalent:

(1) The UC(gln)-module structure on V is inflated from an S(n, r)C-moduleby the map ξ′r,C in (7.1);

(2) V is a quotient module of the UC(gln)-module ΩC(a1)⊗C · · ·⊗CΩC(ar) forsome a ∈ (C∗)r;

(3) V is a quotient module of Ω⊗rC ;

(4) V is a subquotient module of Ω⊗rC .

20 JIE DU

Thus, if V is a simple S(n, r)C-module, then V is finite dimensional by Theorem7.1(2). Hence, V is a quotient module of ΩC(a1)⊗C· · ·⊗CΩC(ar) for some a ∈ (C∗)r,

by the lemma above. Since ΩC(a) is a polynomial representation of UC(gln), by [17,4.3], the tensor product ΩC(a1)⊗C · · · ⊗C ΩC(ar) is a polynomial representation of

UC(gln). Hence, V is also a polynomial representation of UC(gln). We have provedthe following.

Proposition 11.2 ([8, 4.6.4]). Every simple S(n, r)C-module is a polynomial

representation of UC(gln).

We now use Example 10.2(2) to show that every L(Q) with Q ∈ Q(n)r isisomorphic to the inflation of a simple S(n, r)C-module via (7.1).

Let λ = (λ1, . . . , λn) with λj = deg(Qj(u)) and let

Pi(u) =Qi(uq

i−1)

Qi+1(uqi+1), 1 6 i 6 n− 1.

Write Qn(u) = (1− b1u) · · · (1− bλnu) and

Pi(u) =∏

16j6µi

(1− ai,ju),

where µi = λi − λi+1. Let

V = L1 ⊗ · · · ⊗ Ln−1 ⊗Detb1 ⊗ · · · ⊗Detbλn

where Li = Li,ai,1 ⊗ · · · ⊗ Li,ai,µifor 1 6 i 6 n − 1 with Li,ai,j

being an S(n, i)C-

module as defined in Example 10.2(2), and Deta := Ω⊗nC ⊗H(n)C Vs, with s =

(a, aq2, . . . , aq2(n−1)) is the 1-dimensional determinant representation of S(n, n)C(Proposition 10.3). Thus, V is a module for the algebra

A = S(n, 1)⊗µ1 ⊗ · · · ⊗ S(n, n− 1)⊗µn−1 ⊗ S(n, n)

⊗λn .

Since∑

16i6n−1 iµi+nλn =∑

16i6n λi = r, S(n, r)C is a subalgebra of A. Hence,

restriction gives an S(n, r)C-module structure on V .Let wi,ai,k

(resp. vj) be a pseudo-highest weight vector of Li,ai,k(resp., Detbj ),

and let w0 = w1 ⊗w2 ⊗ · · · ⊗wn−1, where wi = wi,ai,1 ⊗wi,ai,2 ⊗ · · · ⊗wi,ai,µi, and

v0 = v1 ⊗ · · · ⊗ vn. Since wi,ai,khas weight (1i, 0n−i) and

Q±j (u)wi,ai,k

=

(1− (ai,kq

i−2j+1u)±1)wi,ai,k, if 1 6 j 6 i;

wi,ai,k, if i < j 6 n,

and

Q±j (u)vi = (1 − (biq

2(n−j)u)±1)vi for 1 6 i 6 λn,

it follows from [17, Lem. 4.1] that

Q±j (u)(w0 ⊗ v0) = Q±

j (u)(w0 ⊗ v0).

Moreover, the weight of w0 ⊗ v0 is

λ = (µ1, 0, . . . , 0) + (µ2, µ2, 0, . . . , 0) + (µn−1, . . . , µn−1, 0) + (λn, . . . , λn).

Let W = S(n, r)C(w0 ⊗ v0) = UC(gln)(w0 ⊗ v0) be the submodule of V gener-ated by w0 ⊗ v0. Then W is a pseudo-highest weight module whose pseudo-highestweight vector is a common eigenvector of ki and Qi,s with eigenvalues zλi and Qi,s,respectively, where Qi,s are the coefficients of Q+

i (u) or Q−i (u). So the simple quo-

tient module of W is isomorphic to L(Q) (cf. the construction in [17, Lem. 4.8]).Hence, we have proved the following.

AFFINE q-SCHUR ALGEBRAS 21

Proposition 11.3 ([8, 4.6.7]). Every L(Q) (Q ∈ Q(n)r) is isomorphic to aninflation of a simple S(n, r)C-module by the map ξ′r,C.

Combining the two propositions above yields the following classification theo-rem.

Theorem 11.4 ([8, 4.6.8]). For any n, r > 1, the setL(Q) | Q ∈ Q(n)r

is a

complete set of nonisomorphic simple S(n, r)C-modules.

It would be interesting to point out that the identification in the n 6 r case (seeTheorem 10.1 for the n > r case) has recently been obtained by Deng–Du (see [18]

for a different proof). Thus, with the notation given in (10.1), we have UC(gln)-

module isomorphisms Ω⊗rC ⊗H(r)C Vs

∼= L(Qs) for all s ∈ S(n)r . In particular, this

gives a bijection from S(n)r to Q(n)r in this case.

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School of Mathematics and Statistics, University of New South Wales, Sydney

2052, Australia.

E-mail address: j.du@unsw.edu.au