The Affineq-Schur Algebra

Journal of Algebra 215, 379᎐411 Ž1999. Article ID jabr.1998.7753, available online at http:rrwww.idealibrary.com on

The Affine q-Schur Algebra R. M. Green* Department of Mathematics and Statistics, Lancaster Uni¨ ersity, Lancaster LA1 4YF, England E-mail: [email protected] Communicated by Gordon James Received March 23, 1998

We introduce an analogue of the q-Schur algebra associated to Coxeter systems of type Aˆny 1 . We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an affine Hecke algebra of type Aˆry 1 , where n G r. This generalizes the original q-Schur algebra as defined by Dipper and James, and the new algebra contains the ordinary q-Schur algebra and the affine Hecke algebra as subalgebras. Using this approach we can prove a double centralizer property. The second construction realizes the affine q-Schur algebra as the faithful quotient of the action of a quantum group on the tensor power of a certain module, analogous to the construction of the ordinary q-Schur algebra as a quotient of UŽ ᒄ ᒉ n .. 䊚 1999 Academic Press

INTRODUCTION The q-Schur algebra, S q Ž n, r ., is a finite dimensional algebra which first appeared in the work of Dipper and James w5, 6x and has applications to the representation theory of the general linear group. The algebra is defined in terms of q-permutation modules for the Hecke algebra arising from the Weyl group of type A. More recently, analogous algebras associated to Weyl groups of type B have been studied Žfor example, w15, 24x., as well as more sophisticated algebras of the same kind w7, 11x which have applications to representation theory. An interesting feature of the q-Schur algebra Žof type A. is that it arises as a quotient of the quantized enveloping algebra UŽ ᒄ ᒉ n .. This relation* The author was supported in part by an E.P.S.R.C. postdoctoral research assistantship. 379 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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ship was explored in w13x. It was shown in w6x that when n G r, the q-Schur algebra S q Ž n, r . is the centralizing partner of the action of the Hecke algebra H Ž Sr . Žwhere Sr is the symmetric group on r letters . on ‘‘q-tensor space.’’ Jimbo w18x established a quantized Weyl reciprocity between the quantized enveloping algebra and the Hecke algebra H Ž Sr . acting on a suitable tensor space. Combining these results means that the faithful quotient of this action of the quantized enveloping algebra is the q-Schur algebra. It is natural to wonder what the corresponding situation is for the Weyl group of type affine A. An affine analogue of Jimbo’s quantized Weyl reciprocity has been described by Chari and Pressley w2x. The significant difference between the situation considered in w2x and the one considered in this paper is that our tensor spaces are infinite dimensional, whereas the one in w2x is finite dimensional. We wish to do this so that the tensor space contains the regular representation of our affine Hecke algebra, which is an infinite-dimensional algebra. Since the representation theory of affine Hecke algebras Žsee w3, 21x. is subtle, our tensor space has an interesting structure as a Hecke algebra module. In particular, it is not completely reducible. The centralizer algebra of this Hecke algebra action on q-tensor space is the affine q-Schur algebra of the title. This is an infinite dimensional associative algebra which contains the affine Hecke algebra and the finite q-Schur algebra as subalgebras, and like the latter algebra, it is defined in terms of q-permutation modules arising from parabolic subalgebras of a suitable Hecke algebra. We also exhibit two bases of this algebra: first a natural one extending the Dipper᎐James basis of the ordinary q-Schur algebra, and second a Kazhdan᎐Lusztig type basis extending Du’s basis w8x for the q-Schur algebra. There is a rival candidate for the title of ‘‘affine q-Schur algebra’’; this arises Žas in the case of the ordinary q-Schur algebra. as a quotient of a mr suitable Hopf algebra acting on a tensor power V$ of a suitable natural module V. We introduce a quantum group, UŽᒄ ᒉ n., and show that the faithful quotient of the action gives an algebra isomorphic to the affine q-Schur algebra of the preceding paragraph. The tensor power V mr is isomorphic to the q-tensor space of the preceding paragraph, although the isomorphism is less easy to describe than in the finite case. The centraliz$ ing algebra of the action UŽᒄ ᒉ n. on V mr is thus isomorphic to the affine Hecke algebra, thus establishing quantized Weyl reciprocity in the affine case. These results raise some interesting questions about canonical bases and $ other properties of UŽᒄ ᒉ n., which we mention at the end of the paper.

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1. WEYL GROUPS AND HECKE ALGEBRAS 1.1. The Affine Weyl Group The Weyl group we consider in this paper is that of type Aˆry1 , where we intend r G 3. This corresponds to the Dynkin diagram in Fig. 1.1. The number of vertices in the graph in Fig. 1.1 is r, as the top vertex Žnumbered r . is regarded as an extra relative to the remainder of the graph, which is a Coxeter graph of type A ry1. We associate a Weyl group, W s W Ž Aˆry1 ., to this Dynkin diagram in the usual way Žas in w16, Sect. 2.1x.. This associates to node i of the graph a generating involution si of W, where si s j s s j si if i and j are not connected in the graph, and si s j si s s j si s j if i and j are connected in the graph. For t g ⺪, it is convenient to denote by t the congruence class of t modulo r, taking values in the set  1, 2, . . . , r 4 . For the purposes of this paper, it is helpful to think of this group as follows. PROPOSITION 1.1.1. There exists a group isomorphism from W to the set of permutations of ⺪ which satisfy the conditions

Ž i q r . w s Ž i. w q r r

for i g ⺪,

Ž 1.

r

Ý Ž t. w s Ý t ts1

Ž 2.

ts1

such that si is mapped to the permutation

¡t ¢t q 1

t ¬~t y 1

if t / i , i q 1, if t s i q 1, if t s i ,

for t g ⺪.

FIG. 1.1. Dynkin diagram of type Aˆry 1.

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Proof. This is given in w22x. For reasons relating to weight spaces which will become clear later, we ˆ of permutations of ⺪. consider a larger group W DEFINITION 1.1.2. Let ␳ be the permutation of ⺪ taking t to t q 1 for ˆ is defined to be the subgroup of permutations of ⺪ all t. Then the group W generated by the group W and ␳ . As will become clear later, the point of ␳ is that conjugation by ␳ will correspond to a graph automorphism of the Dynkin diagram given by rotation by one place.

ˆ to the set of PROPOSITION 1.1.3. There exists a group isomorphism from W permutations of ⺪ which satisfy the conditions Ž i q r . w s Ž i. w q r r

for i g ⺪,

Ž 1.

r

Ý Ž t. w ' Ý t ts1

mod r .

Ž 2.

ts1

Proof. We first observe that Ž2. follows from Ž1.. ŽWe have included Ž2. to emphasize how this result is a generalization of Proposition 1.1.1.. Next, we note that ␳ siq1 ␳y1 s si , where srq1 is interpreted as meaning ˆ is expressible as ␳ z w for some s1. This shows that any element of W z integer z and some w g W. Since ␳ and w both have properties Ž1. and Ž2. given in the statement, it follows that ␳ z w does also. Thus any element ˆ satisfies the properties given. of W Conversely, given a permutation w satisfying Ž1. and Ž2., condition Ž2. shows that r

r

Ý Ž t. w s Ý t ts1

ž /

q kr

ts1

for some integer k. Using the fact that r

r

Ý Ž t. ␳ s Ý t ts1

ž /

q r,

ts1

we see that w␳yk lies in W by Proposition 1.1.1. It follows that w lies in ˆ W.

ˆ is uniquely expressible in the form COROLLARY 1.1.4. Any element of W ␳ w for z g ⺪ and w g W. Con¨ ersely, any element of this form is an element ˆ of W. z

Proof. The second assertion is obvious. The first assertion follows from the uniqueness of the integer k appearing in the proof of Proposition 1.1.3.

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383

ˆ can be expressed in a more familiar way via a semidirect The group W product. PROPOSITION 1.1.5.

ˆ generated by Let S ( Sr be the subgroup of W  s1 , s2 , . . . , sry1 4 .

ˆ consisting of all permutations z satisfying Let Z be the subgroup of W Ž t. z ' t

mod r

ˆ and Wˆ is the semidirect product of S and Z. for all t. Then ⺪ r ( Z eW Proof. It is clear that S and Z are isomorphic to the groups given, ˆ Observe since any permutation z satisfying the given congruence lies in W. also that S l Z s  14 . The other assertions follow easily. It is convenient to extend the usual notion of the length of an element ˆ in the following way. of a Coxeter group to the group W DEFINITION 1.1.6. If w g W, the length, l Ž w . of W is the length of a word of minimal length in the group generators si of W which is equal to w. ˆ Žwhere z g ⺪ The length, l Ž w⬘., of a typical element w⬘ s ␳ z w of W and w g W . is defined to be l Ž w .. 1.2. The Affine Hecke Algebra

ˆ . as in w19x. ŽIt will be shown We now define the Hecke algebra H s H ŽW in Subsection 4.2 that this definition is compatible with the definition given ˆ and in w2x.. The Hecke algebra is a q-analogue of the group algebra of W, is related to W in the same way as the Hecke algebra H Ž Sr . of type A is related to the symmetric group Sr . In particular, one can recover the ˆ by replacing the parameter q occurring in the group algebra of W ˆ . by 1. definition of H ŽW ˆ . over ⺪w q, qy1 x DEFINITION 1.2.1. The affine Hecke algebra H s H ŽW is the associative, unital algebra with algebra generators

 Ts , . . . , Ts 4 j  T␳ , T␳y1 4 1

r

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and relations Ts2 s qTs q Ž q y 1 . , Ts Tt s Tt Ts

Ž 1.

if s and t are not adjacent in the Dynkin diagram,

Ž 2. Ts Tt Ts s Tt Ts Tt

if s and t are adjacent in the Dynkin diagram, Ž 3 .

T␳ Ts iq 1T␳y1 s Ts i .

Ž 4.

In relation Ž4., we interpret srq1 to mean s1. DEFINITION 1.2.2. Let w g W. The element Tw of H ŽW . is defined as Ts i ⭈⭈⭈ Ts i , 1

m

where si1 ⭈⭈⭈ si m is a reduced expression for w Ži.e., one with m minimal.. ŽThis is well-defined by standard properties of Coxeter groups.. ˆ is of form ␳ z w for w g W, we denote by Tw ⬘ the element If w⬘ g W T␳z Tw . ŽThis is well-defined by Corollary 1.1.4.. PROPOSITION 1.2.3.

A free ⺪w q, qy1 x-basis for H is gi¨ en by the set

 Tw : w g Wˆ 4 . Proof. Clearly, H ŽW . is a quotient of the Hecke algebra arising from a Coxeter group of type affine A, and so has a spanning set Tw : w g W 4 . Using relation Ž4. of Definition 1.2.1, one can express any element of H as a sum of elements T␳z T, where T is an element of the subalgebra of ˆ . generated by the elements Ts . It follows that the set given in the H ŽW statement of the proposition is a spanning set. Suppose that there is a linear dependence relation among the elements in the statement. We invoke a standard argument based on Gauss’ Lemma Žsee w15, Theorem 3.2.4x.. By clearing denominators and negative powers of q, we may assume all the coefficients in the dependence relation lie in ⺪w q x and that the largest monomial involved is of minimal possible degree. If the coefficients become identically zero when q is set to 1, then Ž q y 1. divides each coefficient over ⺡Ž q . and therefore over ⺪w q x. This contradicts the minimality assumption, so the coefficients are not identically zero when q is set to 1. However, this gives a dependence relation in the group ˆ a contradiction. algebra of W,

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Remark 1.2.4. The unquantized analogue of the affine Hecke algebra in the sense of w2, Sect. 4.9x is precisely the semidirect product arising in Proposition 1.1.5. It will turn out in Subsection 4.2 that our algebra is the same as the affine Hecke algebra of w2x. It is sometimes useful to know that H can be generated as an algebra more economically as follows. LEMMA 1.2.5.

As a ⺪w q, qy1 x-algebra, H is generated by Ts1, T␳ , and T␳y1 .

Proof. This is obvious from relation Ž4. of Definition 1.2.1. 1.3. Graphs It is sometimes useful to be able to interpret the length of an element in terms of combinatoric properties of graphs. Any permutation of ⺪ can be represented by using the points ⺪ =  0, 14 , where point i in the top row is connected to point Ž i . w in the bottom row by a straight line. Because of condition Ž1. in Proposition 1.1.1 and Proposition 1.1.3, the ˆ have the property that they can permutations representing elements of W be represented on a cylinder with r points evenly spaced on the top and bottom faces, labelled with the congruence classes 1 up to r modulo r. It is clear that one can pass between the cylindrical and planar notation without loss.

ˆ We define ␯ Ž w . to be number of crossDEFINITION 1.3.1. Let w g W. ings in the cylindrical representation of w. The advantage of representing the permutations on a cylinder can be seen from the following result. PROPOSITION 1.3.2.

ˆ Then l Ž w . s ␯ Ž w .. Let w g W.

Proof. Note that ␯ Ž si . s 1 and ␯ Ž ␳ z . s 0. It follows Žby composition of permutations. that ␯ Ž w . F l Ž w . in general. We prove the converse inequality ␯ Ž w . G l Ž w . by induction on ␯ Ž w .. If Ž ␯ w . s 0, one can easily check that Ž t . w s t q z for some z g ⺪, and so w s ␳ z. Now consider the case where the graph of w has a crossing. This means there exist i - j g ⺪ with Ž i . w ) Ž j . w. By consideration of the pairs Ž i, i q 1., Ž i q 1, j y 1., and Ž j y 1, j ., one sees that there exists such a pair i and j for which j s i q 1. Now the element wsi of W⬘ is such that ␯ Ž wsi . s ␯ Ž w . y 1. By induction, wsi is of the length at most ␯ Ž w . y 1, so w s wsi si is of length at most ␯ Ž w ., as desired. This completes the induction. This interpretation of length leads immediately to the following corollary.

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COROLLARY 1.3.3. We ha¨ e

Ž i . w - Ž i q 1 . w m l Ž si w . ) l Ž w . and

Ž i . wy1 - Ž i q 1 . wy1 m l Ž wsi . ) l Ž w . . 1.4. Distinguished Coset Representati¨ es A well-known property of Coxeter groups which is used extensively in the theory of q-Schur algebras is that of distinguished coset representatives with respect to parabolic subgroups. These concepts can be sensibly ˆ Note that Wˆ is not a Coxeter group in an extended to the group W. obvious way, although W is. DEFINITION 1.4.1. Let ⌸ be the set of subsets of S s  s1 , s2 , . . . , sr 4 , ˆ␲ of Wˆ to be excluding S itself. For each ␲ g ⌸, we define the subgroup W  4 Ž that generated by si g ␲ . Such a subgroup is called a parabolic subˆ␲ to emphasize that it is a group.. We will sometimes write W␲ for W subgroup of W. Let ⌸⬘ be the set of elements of ⌸ which omit the generator sr .

ˆ␲ are in fact subgroups of W and Remark 1.4.2. All the subgroups W are parabolic subgroups in the usual sense of Coxeter groups. Furtherˆ␲ is isomorphic to a direct product of Coxeter groups of more, each such W type A Ži.e., finite symmetric groups. corresponding to the connected components of the Dynkin diagram obtained after omitting the elements si which do not occur in ␲ . This assertion is immediate from well-known properties of Coxeter groups. ˆ is the set of those DEFINITION 1.4.3. Let ␲ g ⌸. The subset D␲ of W ˆ elements such that for any w g W␲ and d g D␲ , l Ž wd . s l Ž w . q l Ž d . .

ˆ␲ in W. ˆ We call D␲ the set of distinguished right coset representatives of W The subset D␲y1 is called the set of distinguished left coset representaˆ␲ in W; ˆ elements d g D␲y1 have the property that tives of W l Ž dw . s l Ž d . q l Ž w .

ˆ␲ . for any w g W ˆ Then w s w␲ w ␲ for a PROPOSITION 1.4.4. Let ␲ g ⌸ and w g W. ␲ ˆ unique w␲ g W␲ and w g D␲ .

THE AFFINE q-SCHUR ALGEBRA

387

We ha¨ e w g D␲ if and only if Ž t . w - Ž t q 1. w whene¨ er Ž t, t q 1. ˆ␲ . corresponds to a generator st in W Note. This is a familiar fact for Coxeter groups, but needs to be ˆ checked in the case of the group W. Proof. The second assertion follows from the permutation representaˆ given by Proposition 1.1.3, Corollary 1.3.3, and the fact that Wˆ␲ tion of W is canonically isomorphic to a direct product of finite symmetric groups. ˆ arises as a product Using the same techniques, we see that any w g W ␲ w␲ w such that the lengths add up as claimed. Furthermore, any element ˆ␲ satisfies l Ž w⬘w␲ . s l Ž w⬘. q l Ž w␲ .. It follows that the w⬘w␲ for w⬘ g W set D␲ is indeed a set of right coset representatives. The uniqueness of the decomposition also follows. It is clear that D␲y1 is a set of distinguished left coset representatives with properties analogous to the right coset representatives given by Proposition 1.4.4. PROPOSITION 1.4.5. Let ␲ 1 , ␲ 2 g ⌸. The set D␲ 1 , ␲ 2 [ D␲ 1 l D␲y1 is an 2 ˆ␲ 1 y Wˆ␲ 2-coset representati¨ es, each of irredundantly described set of double W minimal length in its double coset. Proof. This follows from Proposition 1.4.4 using an elementary argument. The double coset representatives D␭, ␮ will play a key role in understanding the structure of the affine q-Schur algebra.

2. THE AFFINE q-SCHUR ALGEBRA AS AN ENDOMORPHISM ALGEBRA 2.1. Construction of the Affine q-Schur Algebra Using the affine Hecke algebra of Section 1, we can now define the affine q-Schur algebra. The definition is designed to be compatible with a quantum group which will be introduced in Section 3, where the motivation for the definitions here will become clearer. The following definition was made in w6x. DEFINITION 2.1.1. A weight is a composition ␭ s Ž ␭1 , ␭2 , . . . , ␭ n . of r into n pieces, that is, a finite sequence of nonnegative integers whose sum is r. ŽThere is no monotonicity assumption on the sequence.. We denote the set of weights by ⌳Ž n, r ..

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R. M. GREEN

The r-tuple l Ž ␭. of a weight ␭ is the weakly increasing sequence of integers where there are ␭ i occurrences of the entry i. ˆ is that subgroup of permutaThe Young subgroup S␭ : Sr : W : W tions of the set  1, 2, . . . , r 4 which leaves invariant the following sets of integers:

 1, 2, . . . , ␭1 4 ,  ␭1 q 1, ␭1 q 2, . . . , ␭1 q ␭2 4 ,  ␭1 q ␭2 q 1, . . . 4 , . . . . The weight ␻ is given by the n-tuple . . . , _^ 1 , 0, 0, ...,_ 0 /. ž^1, 1,` ` ny r

r

Remark 2.1.2. The Young subgroup S␭ : Sr can be thought of as a ˆ␭ for some ␭ g ⌸⬘. Note, however, that different compositions ␭ group W can give rise to canonically isomorphic groups. Also note that the hypothesis n G r is necessary for ␻ to exist.

ˆ␭qt DEFINITION 2.1.3. Let ␭ g ⌸. For t g ⺪, the parabolic subgroup W is the one generated by those elements s iqt where i is such that si lies in ˆ␭. We also use the notation D␭qt with the obvious meaning. W The element x␭qt g H is defined as x␭qt [

Tw .

Ý ˆ␭qt wgW

We will write x␭ for x␭q0 . DEFINITION 2.1.4. The affine q-Schur algebra SˆqŽ n, r . over ⺪w q, qy1 x is defined by Sˆq Ž n, r . [ End H

ž

[

␭ g⌳ Ž n , r .

/

x␭ H ,

ˆ .. where H s H ŽW 2.2. A Basis for Sˆq Ž n, r . One can describe a basis for Sˆq Ž n, r . using techniques familiar from w4x. The main problem is to describe a basis for the Hom space Hom H Ž x␮ H , x␭ H .. LEMMA 2.2.1.

Let ␭ g ⌸⬘. As right H ŽW .-modules,

ˆ. ( x␭ H Ž W

[x tg⺪

␭

␳ tH Ž W . (

[x tg⺪

␭ qt

HŽW . .

THE AFFINE q-SCHUR ALGEBRA

389

Proof. The first isomorphism follows by Proposition 1.2.3 and Corollary 1.1.4. The second isomorphism follows from the definition of x␭qt . Dipper and James remark, at the end of w4, Sect. 2x, that all the results of that section hold for arbitrary finite Coxeter groups. In fact, the results also hold for the infinite Coxeter group W with respect to finite parabolic subgroups, for the same reasons. This proves the following result. LEMMA 2.2.2. The set  ␾␭dqt, ␮ : d g W l D␭qt, ␮ 4 is a ⺪w q, qy1 x-basis for Hom H ŽW . Ž x␮ H Ž W . , x␭qt H Ž W . . , where

␾␭dqt , ␮ Ž x␮ . [

Ý

d⬘g D␯ lW␮

x␭qt Td d⬘ s

Tw

Ý wgW␭qt dW␮

and ␯ is the composition of n corresponding to the standard Young subgroup dy1 W␭qt d l W␮ of W. Using this we can deduce a basis theorem for the affine q-Schur algebra.

ˆ be an element of D␭, ␮ . Write d s ␳ z c DEFINITION 2.2.3. Let d g W Žas in Corollary 1.1.4. with c g W. Then the element ˆ . , x␭ H Ž Wˆ . ␾␭d, ␮ g Hom x␮ H Ž W

ž

/

is defined as

␾␭d, ␮ Ž x␮ . [

Ý

x␭T␳z Tc d⬘

Ý

T␳z x␭qz Tc d⬘ s

d⬘g D␯ lW␮

s

d⬘g D␯ lW␮

Ý

T␳z Tw s

wgW␭qz cW␮

Ý

Tw ,

wgW␭ dW␮

where ␯ is the composition of n corresponding to the standard Young subgroup dy1 W␭qz d l W␮ of W. THEOREM 2.2.4.

A free ⺪w q, qy1 x-basis for Sˆq Ž n, r . is gi¨ en by the set

 ␾␭d, ␮ : ␭ , ␮ g ⌳ Ž n, r . , d g D␭ , ␮ 4 .

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R. M. GREEN

Proof. The proof reduces to showing that a basis for Hom H ŽWˆ .Ž x␮ ˆ ., x␭ H ŽWˆ .. is given by H ŽW

 ␾␭d, ␮ : d g D␭ , ␮ 4 . ˆ . is generated by x␮ , we can use Lemmas 2.2.1 and 2.2.2 to Since x␮ H ŽW reduce the problem to finding a basis for

/ [ Hom

ˆ . , x␭ H Ž Wˆ . s Hom H ŽW . x␮ H Ž W

ž

tg⺪

H ŽW .

Ž x␮ H Ž W . , x␭qt H Ž W . . .

Lemma 2.2.2 gives this as the set

 ␾␭d⬘qt , ␮ : d g W l D␭qt , ␮ , t g ⺪ 4 . But since ␾␭d⬘qt, ␮ is a homomorphism from x␮ to x␭T␳t , ␾␭d, ␮ is a homomorphism from x␮ to x␭ Žset d⬘ s w and t s z .. Furthermore there is a natural bijection between elements of D␭, ␮ and pairs Ž t, W l D␭qt, ␮ . for t g ⺪ given by sending d to Ž z, w . where d s ␳ z w. The proof follows. It is now easy to show that the affine q-Schur algebra contains the affine Hecke algebra and the ordinary q-Schur algebra. PROPOSITION 2.2.5. The set of basis elements

 ␾␭d, ␮ : ␭ , ␮ g ⌳ Ž n, r . , d g Sr l D␭ , ␮ 4 spans a subalgebra of Sˆq Ž n, r . canonically isomorphic to the q-Schur algebra S q Ž n, r .. The set of basis elements

 ␾␻d, ␻ : d g Wˆ 4 ˆ ., where spans a subalgebra canonically isomorphic to the Hecke algebra H ŽW ␾␻d, ␻ is identified with Td . Proof. We recall that the ordinary q-Schur algebra is defined formally in the same way as Sˆq Ž n, r . Žsee w6x., but with H being the Hecke algebra of the symmetric group Sr . The first assertion is now immediate. The second assertion follows from the fact that End H ŽW . Ž x␻ H Ž W . . s End H ŽW . Ž H Ž W . . ( H Ž W . , since H ŽW . is an associative algebra with 1.

THE AFFINE q-SCHUR ALGEBRA

391

2.3. q-Tensor Space and the Double Centralizer Property The definition of the q-Schur algebra in Subsection 2.2 allows the notion of q-tensor space to be introduced in exactly the same way as for the finite q-Schur algebra in w6x. The motivation for the term ‘‘tensor’’ will become clearer in Section 3.

ˆ . with ␾␻d, ␻ g Sˆq Ž n, r ., we DEFINITION 2.3.1. Identifying Td g H ŽW d identify the basis element ␾␭, w of Sˆq Ž n, r . with the coset x␭Td . This makes

[

␭ g⌳ Ž n , r .

x␭ H s² ␾␭d, w : ␭ g ⌳ Ž n, r . , d g D␭:

ˆ . bimodule, which we call q-tensor space, EŽ n, r .. into a Sˆq Ž n, r . y H ŽW LEMMA 2.3.2. ␾␻1 , ␻ .

As a left Sˆq Ž n, r .-module, q-tensor space is generated by

Proof. This is immediate from the fact that ␾␭d, ␻ . ␾␻1 , ␻ s ␾␭d, ␻ . Using the notion of q-tensor space, we can prove that the actions of the affine q-Schur algebra and the affine Hecke algebra on q-tensor space are in reciprocity with each other, that is, that each is the centralizing algebra of the other. This is a straightforward generalization of w6, Theorem 6.6x. THEOREM 2.3.3 ŽThe Double Centralizer Property.

ˆ .. Ži. End Sˆ Ž n, r .Ž EŽ n, r .. s H ŽW q Žii. End H ŽWˆ .Ž EŽ n, r .. s Sˆq Ž n, r .. Proof. The proof of Žii. follows easily from Definition 2.3.1 and the definition of Sˆq Ž n, r .. To prove Ži., first observe that Lemma 2.3.2 implies that any endomorphism in End Sˆq Ž n, r .Ž EŽ n, r .. is determined by its effect on the element x␻ in the subspace x␻ H of q-tensor space. Since

␾␭1, ␭ ␾␮d, ␻ s ␦␭ , ␮ ␾␮d, w , this endomorphism must map x␻ H to itself. The proof of Proposition 2.2.5 shows that these endomorphisms are in canonical correspondence with elements of H ŽW . acting by right multiplication. 2.4. The Kazhdan᎐Lusztig Basis Using the methods of Du w8x, we can describe a second basis for the algebra Sˆq Ž n, r ., analogous to the Kazhdan᎐Lusztig basis for the Hecke algebra of a Coxeter group w20x. Such a basis is called an IC Žintersection

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cohomology. basis; this terminology is justified in the case of the ordinary q-Schur algebra, where the basis has intimate connections to Lusztig’s canonical basis for Uq Žthe ‘‘plus part’’ of a quantized enveloping algebra. and the theory of perverse sheaves Žsee w14, Corollary 4.7x.. The following result is well known for Coxeter groups and their finite parabolic subgroups Žeach of which has a unique longest element..

ˆ The double coset Wˆ␭wWˆ␮ has a LEMMA 2.4.1. Let ␭, ␮ g ⌸ and w g W. unique element of maximal length, denoted by wq. Proof. We may assume that w g D␭, ␮ . Write w s ␳ t w⬘ for w⬘ g W. Then

ˆ␭wWˆ␮ s ␳ t Wˆ␭qt w⬘Wˆ␮ . W ˆ␭qt w⬘Wˆ␮ has a longest Using the result for Coxeter groups, we know that W q element, w⬘ . It is easy to check that the element wqs ␳ t w⬘q satisfies the required criteria. DEFINITION 2.4.2. We denote by F the Žstrong. Bruhat order on a Coxeter group Žsee w16, Sect. 5.9x for the definition.. We extend this order ˆ by stipulating that ␳ a y F ␳ b w Žfor y, w g W . if and only to the group W if a s b and y F w. Let Py, w be the Kazhdan᎐Lusztig polynomial for y, w g W. This polynomial is guaranteed to be zero unless y F w. We extend these polynomials to the group W by defining P␳ a y, ␳ b w [ ␦ a, b Py, w . The indeterminate ¨ is defined to be q 1r2 . ˆ␮ . The element w 0, ␮ is the longest element of W We can now define the Kazhdan᎐Lusztig basis of Sˆq Ž n, r ., taken over the base ring A [ ⺪w ¨ , ¨ y1 x. DEFINITION 2.4.3. Let  ␾␭d, ␮ 4 be the basis of Sˆq Ž n, r . defined in Subsection 2.2. We define

␪␭d, ␮ [ ¨ lŽ w 0 , ␮ .

Ý

␣ z , w ␾␭d, ␮ ,

zg D␭ , ␮

where q

␣ z , w [ ¨ yl Ž w . Pzq , w q . Remark 2.4.4. Note that the sum occurring above is a finite sum, ˆ there is only a finite number of elements y such because for any w g W, that y F w. It is clear that the basis  ␪␭d, ␮ 4 is a free A-basis for Sˆq Ž n, r . since the basis change matrix from the basis  ␾␭d, ␮ 4 is a unit multiple of a unitriangular matrix with respect to a suitable ordering.

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2.5. Generators and Relations For the purposes of Section 3, it is convenient to have a presentation of Sˆq Ž n, r . over ⺡Ž ¨ . via generators and relations. PROPOSITION 2.5.1. The algebra ⺡Ž ¨ . m Sˆq Ž n, r . is generated by elements

 ␾␻d, ␻ : d g Wˆ 4 j  ␾␭1, ␻ : ␭ g ⌳ Ž n, r . 4 j  ␾␻1 , ␭ : ␭ g ⌳ Ž n, r . 4 . The elements ␾␻d, ␻ are subject to the relations of the affine Hecke algebra of Definition 1.2.1 under the identification gi¨ en by Proposition 2.2.5. The generators are also subject to the following relations, where s denotes a ˆ␭ , generator si g W

␾␻1 , ␭ ␾␮1 , ␻ s ␦␭ , ␮

␾␻d, ␻ ,

Ý

Ž 1.

ˆ␭ dgW

␾␻s , ␻ ␾␻1 , ␭ s q␾␻1 , ␭ ,

Ž 2.

␾␭1, ␻ ␾␻s , ␻ s q␾␭1, ␻ .

Ž 3.

Note. Note that s / sr above, since ␭ g ⌸⬘. Proof. Using Proposition 2.2.5 and properties of the ordinary q-Schur algebra Žor the definition of the ␾-basis., it can be verified that the relations given are all true in Sˆq Ž n, r .. Let d s ␳ z c g D␭, ␮ , where c g W Žand thus c g D␭qz, ␮ .. It follows from Definition 2.2.3 that

␾␭1, ␻ ␾␻d, ␻ ␾␻1 , ␮ Ž x␮ . s T␳z x␭qz Tc x␮ s

ž

Ý

d⬘g D␯ lW␮

s P␯ Ž q .

T␳z x␭qz Tc d⬘ x␯

Ý

d⬘g D␯ lW␮

/

T␳z x␭qz Tc d⬘

s P␯ Ž q . . ␾␭d, ␮ , where ␯ is as in Definition 2.2.3 and P␯ is the Poincare ´ polynomial P␯ Ž q . [

Ý

q lŽ w . .

ˆ␯ wgW

This relies on the fact that the expression in parentheses in the second line ˆ␮ , and Wˆ␯ : Wˆ␮ . corresponds to a union of left cosets of W

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It follows that the given set does generate Sˆq Ž n, r ., since P␯ Ž q . is invertible. It also follows that the relations given are sufficient to express the product of two basis elements as a linear combination of others, which completes the proof.

3. THE AFFINE q-SCHUR ALGEBRA AS A HOPF ALGEBRA QUOTIENT The aim of Section 3 is to realize the affine q-Schur algebra as the faithful quotient of the action of a certain quantum group acting on a tensor power of a natural module. This explains why the definition of the q-Schur algebra as given in Section 2 is a natural one. The tensor power of the natural module will be seen to be the analogue of the q-tensor space of Section 2, thus justifying the terminology. ˆ . to be ⺡Ž ¨ . In this section, we take the base ring of Sˆq Ž n, r . and H ŽW by tensoring in the obvious way. $

3.1. The Hopf Algebra UŽᒄ ᒉ n.

$

We now introduce $ a quantum group, UŽᒄ ᒉ n., based on the quantized Ž enveloping algebra U ᒐ ᒉ n. associated to a Dynkin diagram of type Aˆny1. From now on, y will denote congruence modulo n Žas opposed to modulo r ., unless otherwise specified. $ $ The main differences between UŽᒄ ᒉ n. and UŽᒐ ᒉ n. are that the element $ $ Ž . Žsimilar to the K i in UŽᒐ ᒉ n. corresponds to the element K i Ky1 in U ᒄ ᒉ iq1 n $ difference between UŽ ᒄ ᒉ n . and UŽ ᒐ ᒉ n .., and, more strikingly, UŽᒄ ᒉ n. contains a new grouplike element, R, which will play an important role in the results to come. $

DEFINITION 3.1.1. The associative, unital algebra UŽᒄ ᒉ n. over ⺡Ž ¨ . is given by generators y1 Ei , Fi , K i , Ky1 i , R, R

Žwhere 1 F i F n. subject to the relations Ki Kj s Kj Ki ,

Ž 1.

K i Ky1 s Ky1 i i K i s 1, q

K i Ej s ¨ ␧

y

K i Fj s ¨ ␧

Ž 2.

Ž i , j.

Ej K i ,

Ž 3.

Ž i , j.

Fj K i ,

Ž 4.

THE AFFINE q-SCHUR ALGEBRA

Ei Fj y Fj Ei s ␦ i j

y1 K i Ky1 iq1 y K i K iq1

¨ y ¨ y1

Ei Ej s Ej Ei

395 ,

Ž 5.

if i and j are not adjacent,

Ž 6. Fi Fj s Fj Fi

if i and j are not adjacent,

Ž 7. Ei2 Ej y Ž ¨ q ¨ y1 . Ei Ej Ei q Ej Ei2 s 0

if i and j are adjacent,

Ž 8.

Fj2 Fi y Ž ¨ q ¨ y1 . Fj Fi Fj q Fi Fj2 s 0

if i and j are adjacent,

Ž 9.

RRy1 s Ry1 R s 1,

Ž 10 .

Ry1 Kiq1 R s K i ,

Ž 11 .

y1 Ry1 Ky1 iq1 R s K i ,

Ž 12 .

Ry1 Eiq1 R s Ei ,

Ž 13 .

Ry1 Fiq1 R s Fi .

Ž 14 .

By ‘‘adjacent,’’ we mean that i and j index adjacent nodes in the Dynkin diagram. Here,

¡1 ¢0

q

Ž i , j . [~y1

y

Ž i , j . [~y1

␧ and

␧

¡1 ¢0

if j s i; if j s i y 1; otherwise; if j s i y 1; if j s i; otherwise.

Remark 3.1.2. Although it appears at first that the set of defining relations is rather large, it should be noted that in fact the algebra may be y1 generated using only E1 , F1 , K 1 , Ky1 , since the other generators 1 , R, R may be expressed in terms of these by using relations Ž11. ᎐ Ž14.. ŽCompare $ this with Lemma 1.2.5.. Also note that R n lies in the center of UŽᒄ ᒉ n., just ˆ .. as T␳r lies in the center of H ŽW $ It will turn out that the algebra UŽᒄ ᒉ n. can be given the structure of a Hopf algebra Žwhich justifies the use of the term ‘‘quantum group’’.. This is an extension of the well-known Hopf algebra structure w17, Sect. 4.8x of

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the quantized enveloping algebras associated to Kac᎐Moody Lie algebras. Since many of the arguments are familiar, we will highlight only the $ Ž differences between the situation of U s U ᒄ ᒉ n. and quantized enveloping algebras. Although we do not need the antipode in the Hopf algebra for our purposes, we introduce it for completeness. LEMMA 3.1.3. There is a unique homomorphism of unital ⺡Ž ¨ .-algebras ⌬: U ª U m U such that ⌬ Ž 1 . s 1 m 1, ⌬ Ž Ei . s Ei m K i Ky1 iq1 q 1 m Ei , ⌬ Ž Fi . s Ky1 i K iq1 m Fi q Fi m 1, ⌬Ž X . s X m X

y1 for X g  K i , Ky1 4. i , R, R

Note. The elements X above are known as grouplike elements. Proof. This follows by a routine check, which is familiar except for the cases involving the elements R and Ry1 . LEMMA 3.1.4. The map ⌬ is coassociati¨ e. In other words, Ž ⌬ m 1. ⌬ s Ž1 m ⌬ . ⌬. Proof. Since ⌬ is an algebra homomorphism, it suffices to check this on generators. The only unfamiliar case involves R and Ry1 ; in this case, both sides send R to R m R m R and Ry1 to Ry1 m Ry1 m Ry1. LEMMA 3.1.5. There is a unique homomorphism ␧ : U ª ⺡Ž ¨ . of unital ⺡Ž ¨ .-algebras such that

␧ Ž Ei . s ␧ Ž Fi . s 0 and y1 ␧ Ž K i . s ␧ Ž Ky1 . s 1. i . s ␧ Ž R. s ␧ Ž R

Proof. It is trivial to check that ␧ preserves all the relations. LEMMA 3.1.6.

For any u g U, we ha¨ e

Ž ␧ m 1. ⌬ Ž u . s 1 m u and

Ž 1 m ␧ . ⌬ Ž u . s u m 1. Proof. It is enough to verify these claims on the generators. This is easy Žincluding the new cases of u g  R, Ry1 4..

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397

LEMMA 3.1.7. There is a ⺡Ž ¨ .-linear map S: U ª U uniquely determined by the properties S Ž Ei . s yEi Ky1 i K iq1 , S Ž Fi . s yK i Ky1 iq1 Fi , S Ž K i . s Ky1 i , S Ž Ky1 i . s Ki , S Ž R . s Ry1 , S Ž Ry1 . s R, ᭙a, b g U,

S Ž ab . s S Ž b . S Ž a . .

Proof. We verify that S preserves the relations of U, from which it follows that it is determined by its effect on the generators. The only new relations to be checked are relation Ž10. of Definition 3.1.1 Žwhich is clear. and relations Ž11. ᎐ Ž14., which are of the form S Ž Ry1 X iq1 R . s S Ž X i . 4 . Expanding the left hand side, we obtain for X i g  Ei , Fi , K i , Ky1 i S Ž Ry1 X iq1 R . s S Ž R . S Ž X iq1 . S Ž Ry1 . s Ry1 S Ž X iq1 . R. It is then easily verified that Ry1 SŽ X iq1 . R s SŽ X i . in each case. DEFINITION 3.1.8. The ⺡Ž ¨ .-algebra homomorphism ␩ : ⺡Ž ¨ . ª U is defined to send 1 g ⺡Ž ¨ . to 1 g U. We denote the multiplication map in U by ␮. LEMMA 3.1.9. We ha¨ e

␮ Ž S m 1 . ⌬ s ␩␧ and

␮ Ž 1 m S . ⌬ s ␩␧ . Proof. We first check these assertions on the generators. The only new cases involve the elements R and Ry1 , which involve straightforward checks such as

␮ Ž S m 1. ⌬ Ž R . s ␮ Ž S m 1 . Ž R m R . s ␮ Ž Ry1 m R . s 1 s ␩ Ž 1 . s ␩␧ Ž R . .

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This does not complete the proof, because the map S is not an algebra homomorphism, but there is a standard argument Žgiven in w17, Sect. 3.7x. which shows that if the relations in the statement hold on the generators, they hold in general. $

THEOREM 3.1.10. The algebra UŽᒄ ᒉ n. is a Hopf algebra with multiplication ␮ , unit ␩ , comultiplication ⌬, counit ␧ , and antipode S. Proof. This is immediate from Lemmas 3.1.3᎐3.1.7 and 3.1.9. 3.2. Tensor Space Let V be the ⺡Ž ¨ .-vector space with basis  e t : t g ⺪4 . This has a natural $ Ž . U ᒄ ᒉ n -module structure as follows. $

LEMMA 3.2.1. There is a left action of UŽᒄ ᒉ n. on V defined by the conditions Ei e tq1 s e t

if i s t mod n,

Ei e tq1 s 0

if i / t mod n,

Fi e t s e tq1 Fi e t s 0 K i et s ¨ et K i et s et

if i s t mod n, if i / t mod n, if i s t mod n, if i / t mod n,

Re t s e tq1 . Proof. The actions of Ky1 and Ry1 are clear from the above condii tions and relations Ž2. and Ž10. of Definition 3.1.1. One then checks that the other relations are preserved. $

$

Since UŽᒄ ᒉ n. is a Hopf algebra, the tensor product of two UŽᒄ ᒉ n.-mod$ ules has a natural UŽᒄ ᒉ n.-module structure via the comultiplication ⌬. $

DEFINITION 3.2.2. The vector space V mr has a natural UŽᒄ ᒉ n.-module structure given by u.¨ s ⌬Ž u.Ž ry1..¨ . We call this module tensor space. Remark 3.2.3. The definition of Sˆq Ž n, r . in Section 2 is chosen so that the resulting q-tensor space is compatible with the tensor space above. This relationship will be explored later. It is convenient to introduce certain elements of End U Ž V mr ., which we call yi since they are reminiscent of elements of the Hecke algebra in the sense of w2, Sect. 3.1x. This similarity will be explored in Subsection 4.2.

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399

DEFINITION 3.2.4. For each 1 F t F r, we define the invertible linear map yt on V mr by the condition

Ž ej

1

m ⭈⭈⭈ m e j r . . yt [ e j1 m ⭈⭈⭈ m e j ty 1 m eynqj t m e j tq 1 m ⭈⭈⭈ m e j r .

We denote the subspace of V spanned by the elements  e1 , . . . , e n4 by Vn . ŽThis is the vector space which plays the role of V in the case of the ordinary q-Schur algebra.. The following result is a useful tool for reducing problems about Sˆq Ž n, r . to that of the finite q-Schur algebra. PROPOSITION 3.2.5. The elements yi , yy1 lie in End U Ž V mr .. Therefore the i $ action of an element u g UŽᒄ ᒉ n. on tensor space is determined by its action on Vnmr. $

Proof. Since R n lies in the center of UŽᒄ ᒉ n., the linear maps which send e t to e t " n lie in End U Ž V .. Using the comultiplication, this result can be extended to End U Ž V mr ., proving the first assertion. The second assertion holds because V mr s Vnmr.Y, where Y is the algebra of endomorphisms of V mr generated by the yi and their inverses. We require the concept of a weight space of tensor space for the results which follow. This is a straightforward generalization of the situation in finite type A. DEFINITION 3.2.6. The weight ␭ g ⌳Ž n, r . of a basis element e t 1 m e t 2 m ⭈⭈⭈ m e t r of V mr is given by the condition

␭ i [  j: t j ' i mod n4 . The ␭-weight space, V␭, of V mr is the span of all the basis vectors of weight ␭. $

ˆ. 3.3. Relationship between UŽᒄ ᒉ n. and H ŽW As one might expect from the situation in Section 2, the weight space V␻ is of particular importance. In this section we define certain operations on $ this space arising from the action of elements of UŽᒄ ᒉ n.. The motivation for doing this is that these operations will generate an algebra isomorphic ˆ .. This will provide a link with the results of Section 2. to H ŽW DEFINITION 3.3.1. For each 1 F i F r, let ␶ ŽTs i .: V␻ ª V$ ␻ be the endomorphism corresponding to the action of ¨ Fi Ei y 1 g UŽᒄ ᒉ n.. Similarly,

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R. M. GREEN

let ␶ ŽT␳y1 . be the endomorphism corresponding to Fn Fny1 ⭈⭈⭈ Frq1 R , and let ␶ ŽT␳ . be the endomorphism corresponding to Er Erq1 ⭈⭈⭈ Eny1 Ry1 . LEMMA 3.3.2. The endomorphisms ␶ ŽTw . defined abo¨ e satisfy the relations of Definition 1.2.1 Ž after replacing Tw by ␶ ŽTw ... Proof. Using the epimorphism ␣ r : UŽ ᒄ ᒉ n . ¸ S q Ž n, r . described in w13x, one finds that the action of ␶ ŽTs i . on V␻ in the case where i / r corresponds to the action of ␾␻s i, ␻ g S q Ž n, r .. ŽRecall from w13x that S q Ž n, r . is the quotient of UŽ ᒄ ᒉ n . by the annihilator of Vnmr.. This proves relations Ž1. ᎐ Ž3. of Definition 1.2.1 in the case where sr is not involved. It suffices now to prove relation Ž4., from which the other cases of relations Ž1. ᎐ Ž3. may be deduced. The effect of ␶ ŽT␳ . on V␻ is

␶ Ž T␳ . Ž e i1 m ⭈⭈⭈ m e i r . s e j1 m ⭈⭈⭈ m e j1 , where jt s i t y 1 mod r. The effect of ␶ ŽT␳y1 . on V␻ is the inverse of this action. The proof of relation Ž4. now follows by calculation of the action of ¨ Fi Ei y 1 on V␻ using the comultiplication. In the following result, we temporarily replace the ¨ occurring in the definition of the action of ␶ ŽT␻ . by 1, in order to regard the endomorphisms ␶ ŽTw . as endomorphisms of a version of tensor space over ⺡. This modified tensor space is the r-fold tensor power of the ⺡-vector space with basis  e t : t g ⺪4 . We call this process specializing ¨ to 1. LEMMA 3.3.3. Specialize the parameter ¨ to 1. Then the map taking ␶ ŽTw . to w extends uniquely to an isomorphism of algebras between the group algebra ˆ of Wˆ and the algebra ␶ Ž H . generated by the endomorphisms ␶ ŽTw . of ⺡W Definition 3.3.1. Proof. It follows from Lemma 3.3.2 that the algebra generated by the ˆ ␶ ŽTw . is a quotient of the group algebra of W. To complete the proof, it is sufficient Žby Proposition 1.1.3. to show that ˆ corresponds under the isomorphism given to an endomorphism wgW ␶ g ␶ Ž H . such that

␶ Ž e i1 m ⭈⭈⭈ m e i r . s e wŽ i1 . m ⭈⭈⭈ m e wŽ i r . .

ˆ is defined by ␳ Ž t . s t y 1 and s1 acting via the ŽThe left action of W simple transposition Ž1, 2.. It follows from Proposition 1.1.3 that this is faithful..

THE AFFINE q-SCHUR ALGEBRA

401

This claim is easily checked when w is one of the generators s1 , ␳ or ˆ The general case follows by analyzing the action of ␶ ŽTw . on V␻ ␳y1 of W. in the case ¨ s 1. A check shows that this is given by

␶ Ž Tw . Ž e i1 m ⭈⭈⭈ m e i r . s e wŽ i1 . m ⭈⭈⭈ m e wŽ i r . . The proof follows from this. We can obtain an analogue of this result which holds in the quantized case. LEMMA 3.3.4. The map taking ␶ ŽTw . to Tw extends uniquely to an ˆ . and the algebra ␶ Ž H . generated by the isomorphism of algebras between H ŽW endomorphisms ␶ ŽTw . of Definition 3.3.1. Proof. Lemma 3.3.2 shows that the elements ␶ ŽTw . satisfy the required ˆ .. defining relations, so that ␶ ŽTw . is a quotient of H ŽW ˆ Suppose an element of H ŽW . maps to zero under this quotient map. Then by the usual technique of clearing denominators and dividing out by ˆ which maps to zero under the ¨ y 1, we can find an element of ⺡W isomorphism of Lemma 3.3.3. This completes the proof. $

3.4. The Map ␣ r : UŽᒄ ᒉ n. ¸ Sˆq Ž n, r . The results of Subsection 3.3 can be extended to the whole tensor space V mr by using the generators and relations for Sˆq Ž n, r . which were given in Subsection 2.5. This construction relies on the surjective homomorphism ␣ r from UŽ ᒄ ᒉ n . to S q Ž n, r . which was studied in w13x. DEFINITION 3.4.1. For each ␭ g ⌳Ž n, r ., choose 1 E Ž ␭ . g ␣y1 r Ž ␾␭ , ␻ . , 1 F Ž ␭ . g ␣y1 r Ž ␾␻ , ␭ . , 1 G Ž ␭ . g ␣y1 r Ž ␾␭ , ␭ . .

$

We also denote by EŽ ␭. Žrespectively, F Ž ␭., GŽ ␭.. the elements of UŽᒄ ᒉ n. which are the images of EŽ ␭. Žrespectively, F Ž ␭., GŽ ␭.. under the obvious $ Žᒄ ᒉ n.. homomorphism from UŽ ᒄ ᒉ n . to U$ Denote by ␥ the map from UŽᒄ ᒉ n. to EndŽ V mr . corresponding to the action on tensor space arising from the comultiplication. LEMMA 3.4.2. There is a unique homomorphism of algebras

␬ : Sˆq Ž n, r . ª End Ž V mr .

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R. M. GREEN

such that for all ␭ g ⌳Ž n, r .,

␬ Ž ␾␭1, ␻ . s ␥ Ž E Ž ␭ . . , ␬ Ž ␾␻1 , ␭ . s ␥ Ž F Ž ␭ . . , ␬ Ž ␾␭1, ␭ . s ␥ Ž G Ž ␭ . . , ␬ Ž ␾␻s1, ␻ . s ␶ Ž Ts1 . ␥ Ž G Ž ␻ . . , "1

␬ Ž ␾␻␳, ␻ . s ␶ Ž T␳ " 1 . ␥ Ž G Ž ␻ . . . Proof. If such a homomorphism exists, it is unique by Lemma 1.2.5 and Proposition 2.5.1. It remains to check the relations of Proposition 2.5.1. The Hecke algebra style relations hold because of Lemma 3.3.4. The other relations are essentially relations of the ordinary q-Schur algebra. These hold because of Proposition 3.2.5. LEMMA 3.4.3. Let w g Sr , ␭ g ⌳Ž n, r ., and let w s w␭ d be the decomposition of w as in Proposition 1.4.4, so that d g D␭. Consider the basis element e i1 m ⭈⭈⭈ m e i r s eŽ1.w y1 m ⭈⭈⭈ m eŽ r .w y1 of Vnmr ; V mr. Then the action of EŽ ␭. sends this element to one of the form ¨ 2 lŽ w␭.qf Ž ␭ . e j1 m ⭈⭈⭈ m e j r ,

where jt is the i t th component of l Ž ␭. as in Definition 2.1.1, and f is a certain integer-¨ alued function. Proof. This is essentially a statement about the ordinary q-Schur algebra, and is therefore a consequence of w12, Theorem 3.8x. Alternatively, one can argue as follows. The claim holds in the case w s 1 using the definition of q-tensor space Žsee w6, Definition 2.6x and the discussion following it., and the fact that q-tensor space and tensor space are canonically isomorphic Žin the finite dimensional case. by multiplication by a suitable integer power of ¨ Žsee w9, Sect. 1.5x.. Note that this canonical isomorphism respects the notion of weight space Žsee Definition 2.1.1 and Definition 3.2.6.. The general case uses the fact that the actions of H Ž Sr . and UŽ ᒄ ᒉ n . on Vnmr commute, and the definition of the action of H Ž Sr . as given in w9x. Remark 3.4.4. This result determines the action of EŽ ␭. on V␻ because of Proposition 3.2.5.

THE AFFINE q-SCHUR ALGEBRA

403

LEMMA 3.4.5. Consider the basis element e lŽ ␮ .1 m ⭈⭈⭈ m e lŽ ␮ . r of Vnmr ; V mr. Then the action of F Ž ␮ . sends this element to one of the form ¨ gŽ ␮.

Ý

¨ lŽ w . eŽ1.w y1 m ⭈⭈⭈ m eŽ r .w y1 ,

ˆ␮ wgW

where g is a certain integer-¨ alued function. Proof. This is proved using the same techniques as in the proof of Lemma 3.4.3, although this is an easier case. LEMMA 3.4.6. We ha¨ e ␥ ŽU . s imŽ ␬ .. Thus imŽ ␬ . is a quotient of U by the kernel of its action on tensor space. Proof. The second assertion follows if we can establish ␥ ŽU . : imŽ ␬ ., because ␬ Ž ␾ . $ gives the same endomorphism as the action of a suitable Ž element of U ᒄ ᒉ n. for ␾ a generator of Sˆq Ž n, r .. For the first assertion, it is enough to prove that ␥ Ž u. g imŽ ␬ . for y1 u g  E1 , F1 , K 1 , Ky1 4. 1 , R, R

4 holds because of the existence of The case where u g  E1 , F1 , K 1 , Ky1 1 the surjection ␣ r in the finite case and Proposition 3.2.5. We now show that ␥ Ž R . g imŽ ␬ .. It is enough to show that Ž ␥ R .␥ Ž GŽ ␮ .. g imŽ ␬ . for all ␮ g ⌳Ž n, r ., using the decomposition of the identity of Sˆq Ž n, r . into orthogonal idempotents ␾␮1, ␮ . Lemma 3.4.3 and Remark 3.4.4 show that the element ␾␭1, ␻ ␾␻w, ␻ ␾␻1 , ␮ maps under ␬ to a multiple of ␥ Ž R .␥ Ž GŽ ␮ .., where w s ␳y␮ n and where ␭ is such that R sends the weight space ␮ to the weight space ␭. The case for Ry1 is similar but uses w s ␳q␮ 1 . The next result also makes use of specialization. There is no problem in replacing ¨ by 1 here since we are working over an integral form where ¨ y 1 cannot occur as a factor of a denominator. LEMMA 3.4.7. The map ␬ , restricted to the ⺪w q, qy1 x-form of Sˆq Ž n, r . gi¨ en in Theorem 2.2.4, is a monomorphism when the parameter ¨ is specialized to 1. Proof. Using an easy weight space argument, this can be reduced to showing that the images of basis elements ␾␭d, ␮ , for fixed ␭ and ␮ , are independent.

404

R. M. GREEN

Suppose ks␬

ž

Ý dg D␭ , ␮

c d ␾␭d, ␮ s 0.

/

This means that k acts as zero on e␮ [ e lŽ ␮ .1 m ⭈⭈⭈ m e lŽ ␮ . r . Consider the action of one ␾␭d, ␮ which appears in the sum on e␮ . By using the factorization of ␾␭d, ␮ as in the proof of Proposition 2.5.1 together with Lemmas 3.4.3 and 3.4.5 Žin the case ¨ s 1., the isomorphism of Lemma 3.3.3 and the definition of ␬ in Lemma 3.4.2, we have

␾␭d, ␮ .e␮ s

Ý

l w e␭Ž wŽ1.. m ⭈⭈⭈ m e␭Ž wŽ r .. ,

ˆ␮ wgdW

where ␭Ž i . [ l Ž ␭. i , and the scalars l w are positive integers. It follows from this that the actions of ␾␭d, ␮ and ␾␭d⬘, ␮ on e␮ Žwhere d, d⬘ lie in distinct ˆ␭ ᎐Wˆ␮-cosets. are nonzero and have distinct supports. The claim double W follows. We are now ready to show that the two definitions of the affine q-Schur algebra are compatible with each other. THEOREM 3.4.8. O¨ er ⺡Ž ¨ ., the map ␬ gi¨ es an isomorphism between the $ affine q-Schur algebra Sˆq Ž n, r . and the quotient of UŽᒄ ᒉ n. by the kernel of its action on tensor space. Proof. By Lemma 3.4.6, it is enough to show that ␬ is a monomorphism. To do this, we consider an element in ker ␬ and then use the techniques of Lemma 3.3.4 to show that unless this element is zero, we have a counterexample in the case where ¨ is specialized to 1, which contradicts Lemma 3.4.7.

4. CONCLUDING RESULTS 4.1. Comparison of Tensor Space with q-Tensor Space We now show that the q-tensor space of Subsection 2.3 is isomorphic as an Sˆq Ž n, r .-module to tensor space V mr. In Section 4, we continue to use ⺡Ž ¨ . as a base ring unless otherwise stated. LEMMA 4.1.1. ¨ ector

ˆ for w g W.

$

As a UŽᒄ ᒉ n.-module, tensor space V mr is generated by any e w s e wŽ1. m ⭈⭈⭈ m e wŽ r .

THE AFFINE q-SCHUR ALGEBRA

405

Proof. Using Lemma 3.4.3, it is enough to show that V␻ is generated as $ Ž . . of V␻ a U ᒄ ᒉ n -module by e w . We introduce endomorphisms ␶ ŽTsy1 i $ y1 corresponding to the action of the elements ¨ Ei Fi y 1 of UŽᒄ ᒉ n. on V␻ . . is the inverse of ␶ ŽTs .. It is easily checked that ␶ ŽTsy1 i i Let e j1 m ⭈⭈⭈ m e j r be a basis element of V␻ . Let 1 F pŽ j, i . F r be the unique integer such that j pŽ j, i. ' i mod n. It is not hard to show that if pŽ j, i . - pŽ j, i q 1. then

␶ Ž Ts i .Ž e j1 m ⭈⭈⭈ m e j r . s ¨ U e s i Ž j1 . m ⭈⭈⭈ m e s i Ž j r . , for some integer ), and if pŽ j, i . ) pŽ j, i q 1. Žthe only other possibility., we have

␶ Ž Tsy1 . Ž e j1 m ⭈⭈⭈ m e jr . s ¨ U e s iŽ j1 . m ⭈⭈⭈ m e s iŽ jr . . i If w s ␳ " 1, we have

␶ Ž Tw . Ž e j1 m ⭈⭈⭈ m e j r . s ¨ U e wŽ j1 . m ⭈⭈⭈ m e wŽ j r . . We can now use the techniques of Lemma 3.3.3 to deduce the claim. THEOREM 4.1.2. As Sˆq Ž n, r .-modules, q-tensor space and tensor space are isomorphic. The isomorphism may be chosen to take ␾␻1 , ␻ to e␻ s e1 m ⭈⭈⭈ m e r . Proof. By Lemma 4.1.1, we need only show that no element of q-tensor space annihilates the vector e␻ . Suppose ks

Ý

␭ g⌳ Ž n , r . , dg D␭

c␭ , d ␾␭d, ␻

annihilates e␻ . We may assume that only one ␭ occurs in the sum. In this case, k s ␾␭1, ␻ .

žÝ

c␭ , d ␾␻d, ␻ ,

dg D␭

/

and if k acts as zero, so does

Ý Ý

c␭ , d ␾␻w,d␻ .

ˆ␭ dg D␭ wgW

ŽTo see the latter assertion, premultiply by ␾␻1 , ␭ and use Lemma 3.4.5.. This contradicts Lemma 3.3.4 unless all the c␭, d are zero.

406

R. M. GREEN

Remark 4.1.3. The isomorphism between q-tensor space and tensor space is not a natural one as in the case of the ordinary q-Schur algebra or the case q s 1. To see this, compare the actions of ␾␻s r, ␻ on the generating vector: in the case of q-tensor space, this produces a combination of one other basis vector, but in the case of tensor space, e1 occurs to the left of e r in e␻ , so a combination of two basis vectors is produced. 4.2. Presentations for the Affine Hecke Algebra An important consequence of Theorem 4.1.2 is that it provides, in conjunction with Theorem 2.3.3, a right action of H on tensor space, and $ furthermore, the centralizing algebra of the action of UŽᒄ ᒉ n. on tensor space is isomorphic to H . This means that the endomorphisms yi of Definition 3.2.4 can be regarded as elements of H . The results of this section consider this action further to show that our affine Hecke algebra is in fact isomorphic to the affine Hecke algebra in the sense of w2, 3.1 Definitionx. Let

LEMMA 4.2.1.

e j1 m ⭈⭈⭈ m e j r be a basis element of V␻ satisfying 1 F jt F r for all t. Suppose that ji s r y 1 and jiq1 s r. Then we ha¨ e

Ž ej

1

m ⭈⭈⭈ m e j r . .Ts i yi Ts i s Ž e j1 m ⭈⭈⭈ m e j r . .¨ 2 yiq1 .

Proof. By definition of the endomorphism yiq1 in Definition 3.2.4, the right hand side is easily seen to be ¨ 2 e j1 m ⭈⭈⭈ m e j i m eynqj iq 1 m e j iq 2 m ⭈⭈⭈ m e j r .

Because of the way S q Ž n, r . embeds in Sˆq Ž n, r ., the action of Ts i on Vnmr is the same as in the finite case. We therefore use the definition of this action in w9, Sect. 1.5x to show that

Ž ej

1

m ⭈⭈⭈ m e j r . .Ts i yi s ¨ e j1 m ⭈⭈⭈ m eynqj iq 1 m e j i m ⭈⭈⭈ m e j r .

It suffices to show that

Ž ej

1

m ⭈⭈⭈ m eynqj iq 1 m e j i m ⭈⭈⭈ m e j r . .Ts i s ¨ Ž e j1 m ⭈⭈⭈ m e j i m eynqj iq 1 m ⭈⭈⭈ m e j r . .

Recall that yn q jiq1 s yn q r. There are no other indices with values between yn q r y 1 and 0.

THE AFFINE q-SCHUR ALGEBRA

407

It is enough to establish

Ž eŽ nyrq1.qj

1

m ⭈⭈⭈ m e1yrqj iq 1 m eŽ nyrq1.qj i m ⭈⭈⭈ m eŽ nyrq1.qj r . .Ts i

s ¨ Ž eŽ nyrq1.qj1 m ⭈⭈⭈ m eŽ nyrq1.qj i m e1yrqj iq 1 m ⭈⭈⭈ m eŽ nyrq1.qj r . , and then act $

Ryn qry1 g U Ž ᒄ ᒉ n . on the left of both sides. Since all the indices lie between 1 and n and n s Ž n y r q 1 . q ji ) 1 y r q jiq1 s 1, the claim follows from the action of H on Vnmr. COROLLARY 4.2.2. We ha¨ e Ts i yi Ts i s ¨ 2 yiq1. Proof. This follows from Lemma 4.1.1, Lemma 4.2.1, and the double $ centralizer property for UŽᒄ ᒉ n. and H on tensor space. Suppose 1 F i - r, 1 F j F r, and j / i, i q 1. Let

LEMMA 4.2.3.

e k 1 m ⭈⭈⭈ m e k r be a basis element of V␻ satisfying 1 F k t F r for all t. Suppose that k i s r y 2, k iq1 s r y 1, and k j s r. Then we ha¨ e

Ž ek

1

m ⭈⭈⭈ m e k r . . y j Ts i s Ž e k 1 m ⭈⭈⭈ m e k r . .Ts i y j .

Proof. This is proved using an argument very similar to that in the proof of Lemma 4.2.1. COROLLARY 4.2.4. We ha¨ e y j Ts i s Ts i y j if j / i, i q 1. Proof. This uses Lemma 4.1.1, Lemma 4.2.3, and the double centralizer property. We can now establish an isomorphism between H and the affine Hecke algebra of w2, Sect. 3x.

ˆ . is isomorphic to the algebra gi¨ en by THEOREM 4.2.5. The algebra H ŽW generators

 ␴i " 1 : 1 F i - r 4 j  yi" 1 : 1 F i F r 4

408

R. M. GREEN

and defining relations

␴i ␴y1 s ␴y1 i i ␴i s 1,

Ž 1.

␴i ␴iq1 ␴i s ␴iq1 ␴i ␴iq1 ,

Ž 2.

␴i ␴j s ␴j ␴i

if < i y j < ) 1,

Ž ␴i q 1 . Ž ␴i y ¨ 2 . s 0,

Ž 4.

y j yy1 s yy1 j j y j s 1, yj yk s yk yj , y j ␴i s ␴i y j

Ž 3.

Ž 5. Ž 6.

if j / i , i q 1,

␴i yi ␴i s ¨ 2 yiq1 .

Ž 7. Ž 8.

The isomorphism may be chosen to identify ␴i with Ts i . Proof. We identify ␴i with Ts i . Relations Ž1. ᎐ Ž4. are known from Definition 1.2.1. Relations Ž5. and Ž6. are obvious from considering the faithful action of H on tensor space. Relations Ž7. and Ž8. follow from Corollaries 4.2.4 and 4.2.2, respectively. These relations are sufficient since w2, 3.1 Lemmax shows that a basis for the algebra given by these generators and relations has a basis y 1c1 ⭈⭈⭈ yrc r Tw as c i g ⺪ and Tw ranges over a basis for H Ž Sr .. Since the algebra in the statement is clearly isomorphic to its own opposite algebra, there is also a basis of the form Tw y 1c1 ⭈⭈⭈ yrc r . It is not hard to verify that the map Tw y 1c1 ⭈⭈⭈ yrc r ¬ e␻ .Tw y 1c1 ⭈⭈⭈ yrc r is injective, from which we deduce that the algebra of w2x acts faithfully on tensor space under the identifications given. It follows that the relations given in the statement are sufficient. Remark 4.2.6. Using Theorem 4.2.5 we can give a third construction of the affine q-Schur algebra more compatible with the results of w2x. This is formally the same as Definition 2.1.4, but uses the version of the affine Hecke algebra given in w2x. The presentation for H just given is more convenient when dealing with tensor space, in contrast to Proposition 1.2.3 which is adapted for q-tensor

THE AFFINE q-SCHUR ALGEBRA

409

space. Using the basis Tw ⌸ yic i of H which was mentioned in the proof of the theorem, one can explicitly calculate the action of the generators Ts i on tensor space. We do not do this here, because it is much more complicated than in the case of type finite A. In type finite A, the action of Ts i on a basis element of Vnmr produces a linear combination of at most two basis elements, but in the case of the action of Ts i on V mr in type affine A, it is possible for arbitrarily many basis elements to be involved. This happens because we are using a basis of H whose structure constants are much more complicated than those of the basis given in Proposition 1.2.3. ŽThis complication arises from relation Ž8. in the statement of the theorem.. 4.3. Some Questions $

It seems from the results of Section 3 that the Hopf algebra UŽᒄ ᒉ n. is a natural object. Further evidence for this is provided by the results of w25, Sect. 11x, where it is proved that the affine q-Schur algebra introduced in this paper occurs naturally in a K-theoretic context. This generalizes the interpretation in w1x of the ordinary q-Schur algebra in terms of the geometry of the action of the general linear group Žover a finite field. acting on certain pairs of flags. . and the Comparing the situation with the relationship between UŽ ᒄ ᒉ n$ ordinary q-Schur algebra raises some more questions about UŽᒄ ᒉ n., such as the following ones. Q UESTION 4.3.1. Can one describe a suitable A-form of the algebra $ Ž U ᒄ ᒉ n. which maps under ␣ r to the natural A-form of Sˆq Ž n, r .? This is an important question because it would show that our results also hold at roots of unity. QUESTION 4.3.2. To what extent do these results hold in the case where n - r? In this case, there is no weight ␻ , and one should not expect the Hecke algebra to act faithfully on tensor space, since it does not act faithfully in the case of the ordinary q-Schur algebra. In the latter case, it can be shown that the centralizing algebra of the q-Schur algebra is a quotient of the Hecke algebra. Something similar may hold in the affine case. A main obstacle to proving this lies in the difficulty of embedding Sˆq Ž n, r . in Sˆq Ž n q 1, r .. This is in contrast to the finite case, where there are obvious embeddings. The lack of an obvious determinant map also means that it is not clear how to realize Sˆq Ž n, r . as a quotient of Sˆq Ž n, r q n..

410

R. M. GREEN

$

QUESTION 4.3.3. Is there a basis for UŽᒄ ᒉ n. similar to Lusztig’s canonical basis for the modified quantized en¨ eloping algebras U˙ which is compatible with the Kazhdan᎐Lusztig basis of Subsection 2.4 ¨ ia the maps ␣ r? Lusztig’s algebras U˙ are defined in w23, Sect. 23x. They are equipped with ˙ generalizing the canonical bases for the algebras Uq canonical bases B, Žwhich are the subalgebras of quantized enveloping algebras generated by the elements Ei .. In the case of the ordinary q-Schur algebra, these bases have been shown in w10x to be compatible with the Kazhdan᎐Lusztig bases for the q-Schur algebra, in the sense that the basis elements B˙ map either to zero or to Kazhdan᎐Lusztig basis elements. It seems reasonable to hope therefore that an analogous result exists for the affine case. QUESTION 4.3.4. Do the structure constants for the affine q-Schur algebra with respect to the Kazhdan᎐Lusztig basis lie in ⺞w ¨ , ¨ y1 x? This property is true for the ordinary q-Schur algebra, as was proved in w14x. If the more general result is true, it should have an interpretation in terms of intersection cohomology.

ACKNOWLEDGMENTS The author thanks J. Du and R. J. Marsh for helpful conversations.

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