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On Endomorphism Algebras Arising from Multiparameter Hecke Algebras
195, 308]319 Ž1997. JA977032
JOURNAL OF ALGEBRA ARTICLE NO.
On Endomorphism Algebras Arising from Multiparameter Hecke Algebras Hebing Rui*, † Department of Mathematics, East China Uni¨ ersity of Technology, Shanghai, 200093, People’s Republic of China Communicated by Gordon James Received November 7, 1996
INTRODUCTION Let GŽ q .4 be a family of finite groups of Lie type having ŽW, S . as its irreducible Coxeter system. For each parameter q, let c s be the integer satisfying < B Ž q .: sB Ž q . l B Ž q .< s q c s for each s g S, where B Ž q . is a fixed Borel subgroup of GŽ q .. Then c s s c t if s and t are W-conjugate. Let Zw q 1r2 , qy1r2 x be the ring of Laurent polynomials in indeterminate q 1r2 . Then the Hecke algebra H corresponding to GŽ q . over A has a standard basis Tw < w g W 4 satisfying the conditions
Ž Ts y q c . Ž Ts q 1 . s 0,
if s g S,
Tx Ty s Tx y ,
if l Ž xy . s l Ž x . q l Ž y . ,
s
where l Ž . is the length function on W. For any I ; S, let x I s Ý w g W I Tw , where WI is the standard parabolic subgroup of W generated by I. Let A s End H Ž[I ; S x I H .. If GŽ q . is the general linear group GLŽ n, q ., then A is known as a q-Schur algebra Žsee wDJ1, DJ2x.. In wPWx, q-Schur algebras have been proved to be quasihereditary algebras in the sense of wCPS1x. In other words, the category of left A-modules is a highest weight category in the sense of wloc. cit.x. * The author gratefully acknowledges support by the National Natural Science Foundation 19501016 in China and Large ARC Grant A69530243 from the Australian Research Council. † E-mail: [email protected]. 308 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
ENDOMORPHISM ALGEBRAS
309
In this paper, we consider the cases when G is either U2 nŽ q . or U2 nq1Ž q ., where UnŽ q . is the group of unitary n by n matrices over the field of q elements. The Weyl group W associated to G is of type Bn . The Dynkin diagram of W is as follows: 1
`
2 n } ` } ??? s ` .
Let c s be the integers associated to G. If G s U2 nŽ q ., then cs s
½
1, 2,
if s s sn , otherwise.
cs s
½
3, 2,
if s s sn , otherwise.
If G s U2 nq1Ž q ., then
Let H be the Hecke algebra corresponding to either U2 nŽ q . or U2 nq1Ž q .. Let A be the endomorphism algebra associated to H . In this paper, we study the representation theory of endomorphism algebra A by using Kazhdan]Lusztig theory. Our method is based on the representation theory of standardly based algebras given in wDR1x. The main result of this paper is that the Kazhdan]Lusztig basis of A is a standard base in the sense of wloc. cit., 1.2.1x Žsee Ž2.1... Using the results in wloc. cit.x, we describe the simple modules and standard modules in the category of left A-modules. Our result shows that the category of left A-modules has results similar to the case of a highest weight category Žsee Ž4.4... When we specialize q to 1, we get the results on endomorphism algebras End kW Ž[J : S TJ . where W is the Weyl group of type Bn and TJ is the permutation module on the cosets WJ w4w g W . The contents of this paper are organized as follows. In Section 1, we recall some basic definitions and results. Let H be the Hecke algebra associated to U2 nŽ q . or U2 nq1Ž q .. In Section 2, we prove that the Kazhdan]Lusztig basis of H is a standard base by a result due to Graham in wGrax. In Section 3, we prove the Kazhdan]Lusztig basis of A is a standard base. In Section 4, we give some results on the representation theory of endomorphism algebra A.
1. HECKE ALGEBRAS Let W be a finite Coxeter group with S as its distinguished generator set of simple reflections. Let A s Zw t, ty1 x be the ring of Laurent polyno-
310
HEBING RUI
mials in indeterminate t. For any s g S, let c s be a positive integer such that c s s c t if s and t are W-conjugate. Then the generic Hecke algebra H associated to W is an associative algebra over A with free A-basis Tw < w g W 4 . Let l Ž . be the length function on W. Then,
Ž Ts y t 2 c . Ž Ts q 1. s 0,
if s g S,
Tx Ty s Tx y ,
if l Ž xy . s l Ž x . q l Ž y . .
s
Ž 1.1.
If the parameters c s are determined by finite groups GŽ q . and W the Weyl group associated to GŽ q ., we say the generic Hecke algebra H is associated to GŽ q .. Ž1.2. Let F be the Bruhat]Chevalley order on W. In wLu1x, Lusztig introduced another basis Bw < w g W 4 for H such that Bw s
Ý
PyU, w Ty ,
yFw
where Py,U w is known as the Kazhdan]Lusztig polynomial Žresp. generalized Kazhdan]Lusztig polynomial. if H has equal parameters Žresp. unequal parameters.. Ž1.3. Cells. Let FL be the preorder on W such that x FL y is there is z g W such that B x appears in the expression of Bz B y with non-zero coefficient. Let FR be the preorder on W such that x FR y if xy1 FL yy1 . Let FL R be the preorder on W generated by FL and FR. The equivalent relations corresponding to x g L, R, LR4 are denoted by ;x and the equivalent classes corresponding to L, R, LR are called left cells, right cells, and two-sided cells Žresp. generalized left, right and two-sided cells if H has unequal parameters.. Ž1.4. Let H be the generic Hecke algebra associated to finite groups U2 nŽ q . or U2 nq1Ž q .. Then H is of type Bn Žor Cn . with unequal parameters Žsee Introduction.. H is isomorphic to a subalgebra of a Hecke algebra associated to a certain symmetry group. In fact, if the Hecke algebra H is associated to U2 nŽ q ., then we consider the Hecke algebra associated to the symmetry groups S 2 n . For 1 F i F 2 n y 1, let t i s Ž i, i q 1. be the basic transposition. Then T s t i < 1 F i F 2 n y 14 is the distinguished generator set of S 2 n . Let a be the non-trivial isomorphism of S 2 n such that a ŽT . s T. Then the subgroup S 2an s w g S 2 n < a Ž w . s w4 is isomorphic to the Weyl group of type Bn with U s u i < 1 F i F n, u i s t i t 2 nyi if 1 F i F n y 1 and u n s t n 4 as its distinguished generator set of simple reflections. Let c i [ c u i s l Ž u i . where l Ž . is the length function on S 2 n . Let H Ž S 2an . be the Hecke
311
ENDOMORPHISM ALGEBRAS
algebra associated to S 2an with respect to c i . Then H Ž S 2an . ( H ,
Ž 1.5.
where H is the Hecke algebra associated to U2 nŽ q .. If H is associated to the finite group U2 nq1Ž q ., then H is isomorphic to a subalgebra of the Hecke algebra associated to the symmetry group S 2 nq1. Hereafter, we deal with the Hecke algebra with parameters associated to U2 nŽ q .. The second case can be dealt with similarly. Let W Ž Bn . be the Weyl group of type Bn with S s s1 , s2 , . . . , sn4 as its distinguished generator set of simple reflections. Let c : W Ž Bn . ª S 2an be such that c Ž si . s u i for 1 F i F n. Then c is an isomorphism. By Ž1.5., we have, for any x, y g W Ž Bn ., x ;L y if and only if c Ž x . ;L c Ž y . in S 2an . The last condition implies c Ž x . ;L c Ž y . in S 2 n by the result due to Lusztig in wLu1, Sect. 9.2x Žsee also wBre, Sect. 4.3x.. From here onwards, we consider w g W Ž Bn . as w g S 2an by abusing notation. 2. STANDARDLY BASED ALGEBRAS In this section, we prove the Kazhdan]Lusztig basis for Hecke algebras associated to U2 nŽ q . or U2 nq1Ž q . is a standard base introduced in wDR1x. First, we recall the definition of standard base. Ž2.1. Assume that R is a commutative ring with unit 1. Let A be an R-algebra and Ž L, F. a poset. A is called a standardly based algebra on L if the following conditions hold. Ža. For any l g L, there are index sets I Ž l. and J Ž l. such that B l s a il, j Ž i , j . g I Ž l . = J Ž l . .
½
5
Žb. The disjoint union B s Dlg L B l is a free R-basis of A. Žc. For any a g A, a i,l j g B, we have
Ž 2.2.
a ? a il, j '
Ý i9gI Ž l .
a il, j ? a '
Ý j9gJ Ž l .
l )l f i9, l Ž a, i . a i9, . j mod Ž A
fl , j9 Ž j, a . a il, j9 modŽ A ) l . ,
where A ) l is the R-submodule of A spanned by B m with m ) l and f i9, lŽ a, i ., fl, j9Ž j, a. g R are independent of j and i, respectively. Such a base B is said to be a standard base for the standardly based algebra A.
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HEBING RUI
Standardly based algebras are generalized versions of cellular algebras introduced by Graham and Lehrer in wGLx. Ž2.3. A composition l of a non-negative integer n is a sequence Ž l1 , l 2 , . . . , . such that Ý`is1 l i s n. If l i G l j for all i F j, then l is said to be a partition. For positive integer n, r, let LŽ n, r . Žresp. LqŽ n, r .. be the set of all compositions Žresp. partitions. of r with at most n parts. For l g LqŽ n, r ., we may identify l with its corresponding diagram which consists of crosses arranged in a manner as illustrated by the example l s Ž421. g LqŽ 3, 7. for which we have = ls = =
= =
=
=
A l-tableau t is obtained by replacing each cross by one of the numbers 1, 2, . . . , r. The symmetric group W s S r acts on the set of l-tableaux by letter permutations. For any partition l of r, we consider l-tableaux such that the number of entries i is equal to m i , where m is a composition of r. Such tableaux are called l-tableaux of type m. Let T Ž l, m . denote the set of all l-tableaux of type m. A l-tableau t is called semistandard Žresp. standard. if its entries are weakly increasing Žresp. strictly increasing . along each row and increasing along each column. Let T0 Ž l, m . be the set of all semistandard l-tableaux of type m. If n G r, we set v s Ž1r .. Clearly, T0 Ž l, v . is the set of all standard l-tableaux. Ž2.4. Robinson]Schensted Map. Let W be the symmetry group S r . Let f : w ¬ ŽtŽ w ., sŽ w .. be the Robinson]Schensted map from W to the set of pairs of standard tableaux Žsee, eg., wBx.. In wBVx, Barbasch and Vogan proved that, for x, y g W,
Ž 2.5.
x ;L y Ž resp. x ;R y . if and only if s Ž x . s s Ž y .
Ž resp. t Ž x . s t Ž y . . . Ž2.6. Let a be the non-trivial automorphism on S 2 n preserving the generator set T Žsee Ž1.4... In wLu1x and wBre, Sect. 4.3x, Lusztig and Bremke proved that any generalized left cell G of S 2an satisfies G s G1a for some left cell G1 of S 2 n . Let L be the set of generalized two-sided cells of W Ž Bn .. For any V g L, let I Ž V . s t g T0 Ž l , v . t s t Ž w . for some w g V 4 J Ž V . s s g T0 Ž l , v . s s s Ž w . for some w g V 4 , where l is the two-sided cell of S 2 n containing V. ŽHere, we consider V as a two-sided cell of S 2an .. By the result due to Bremke and Lusztig Žsee
ENDOMORPHISM ALGEBRAS
313
wLu1, Theorem 11x or wBre, P. 63, line 3x., l a is a union of generalized two-sided cells of S 2an . In fact, this result can be deduced from wloc. cit. 4.3.8x.. Obviously, I Ž V . s J Ž V . for a generalized two-sided cell of W Ž Bn . since tŽ w . s sŽ wy1 . and sŽ w . s tŽ wy1 . for any w g S 2 n . I Ž V . is a subset of T0a Ž l, v .; the latter consists of the elements in T0 Ž l, v . which are fixed by an evacuation map defined by Schutzenberger in wSx. Such elements are ¨ called symmetric standard tableaux Žsee wGra, Chap. 2x.. The following result follows from a result due to Graham wloc. cit. Chap. 3x. Ž2.7. THEOREM. Let H be the Hecke algebra associated to finite groups U2 nŽ q . or U2 nq1Ž q .. Then the Kazhdan]Lusztig basis of H is a standard base in the sense of Ž2.1.. Proof. For any w g W Ž Bn ., let Bw [ BPl, Q where l is the generalized two-sided cell of W Ž Bn . containing w and Ž P, Q . s f Ž w . g I Ž l. = I Ž l.. Let L be the set of all generalized two-sided cells of W Ž Bn .. Then L is a poset with partial order F , where F is the reversed order of FL R . For any l g L, let H ) l be the free A-submodule of H spanned by BPm, Q with P, Q g I Ž m . and m ) l. For all h g H Ž Bn ., we have
Ž 2.8.
hBPl, Q '
Ý f P 9l Ž h, P . BPl9, Q mod H ) l
BPl, Q h '
Ý fl , Q9 Ž Q, h . BPl, Q9 mod H ) l
by Ž2.5. and Corollary Ž4.3.10. in wBrex Žthe latter implies that if z9 FL z and z ;L R z9, then z ;L z9.. Moreover, f P 9 lŽ h, P . and fl , Q9Ž Q, l. are elements in A, which are independent of Q and P, respectively, by Ž1.5. and wGra, 3.2x Žthe latter says that the left cell modules EŽ G1 . and EŽ G2 . are isomorphic if G1 and G2 are in the same generalized two-sided cell of H Ž S 2an ... Thus the Kazhdan]Lusztig basis of H is a standard base in the sense of Ž2.1.. Let R be a commutative noetherian ring over A with unit 1. Let q 1r2 g R be the image of t. Let HR s H mA R. Then HR is the Hecke algebra over the commutative noetherian ring R. We denote Tw and Bw by Tw m 1 and Bw m 1 by abusing notation. By Theorem Ž2.7., we have the following result. Ž2.9. COROLLARY. Let HR be the Hecke algebra associated to one of the finite groups U2 nŽ q . or U2 nq1Ž q .. Then the Kazhdan]Lusztig basis Bw < w g W Ž Bn .4 is a standard base of HR in the sense of Ž2.1..
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HEBING RUI
3. ENDOMORPHISM ALGEBRAS End H Ž[J : S x J H . In this section, we prove our main result, which says the KzahdanLusztig basis of End H Ž[J : S x J H . is a standard base if H is the Hecke algebra associated to U2 nŽ q . or U2 nq1Ž q .. At first, we introduce the Kazhdan]Lusztig basis for End H Ž[J : S x J H .. When H has equal parameters, the Kazhdan]Lusztig basis for End H Ž[J : S x J H . has been introduced by Du in wDu1x. Let W be a finite Coxeter group of non-simply laced type. Let S be its distinguished generator set of simple reflections. For each subset I ; S, let WI be the parabolic subgroup of W generated by I, and DI the set of representatives of minimal length in the right cosets WI R W. Thus, the set DI J s DI l Dy1 for I, J ; S consists of the representatives of minimal J length in the double cosets WI R WrWJ . Let Dq I J be the set of the representatives of maximal length in the double cosets WI R WrWJ . Let G be an abelian group. We now assume that a total order e on G is given, which is compatible with the structure of G. For any s g S, we associate a parameter q s g G such that q s s qt if s, t are W-conjugate. We also assume that q s g Gq for all s g S, where Gq is the set of elements which are strictly positive for the total order e. Let H be the Hecke algebra associated to W over Zw G x. Then H has a free Zw G x-basis Tw < w g W 4 such that
Ž Ts y qs . Ž Ts q 1 . s 0,
if s g S,
Tx Ty s Tx y ,
if l Ž xy . s l Ž x . q l Ž y . .
If G s ² t 1r2 :, and q s s t 2 c s for some integer c s satisfying c s s c t if s and t are W-conjugate, then the Hecke algebra defined above coincides with that defined in Ž1.1.. Let Bwe < w g W 4 be the Kazhdan]Lusztig basis with respect to the total order e Žsee wLu1x.. Hereafter, we use Bw to replace Bwe by abusing notation. For any D ; W, let TD s Ý x g D Tx . For any I, J : S, define HI J to be the A-free module spanned by the elements TD , D g WI R WrWJ . A direct computation shows that HI J can be characterized as follows: HI J s h g H < Ts h s qs h, hTt s qt h, for all s g I and t g J 4 . Ž3.1. PROPOSITION ŽCompare wCur, 1.10x.. Let H be the Hecke algebra o¨ er Zw G x, where G is an abelian group. Let e be a total order on G. Then Kazhdan]Lusztig base elements Bw 4w g D qI J with respect to e are a free Zw G x-basis for HI J .
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ENDOMORPHISM ALGEBRAS
Proof. This follows from an argument similar to wCur, 1.10x. Now we introduce the Kazhdan ] Lusztig basis for A [ End H Ž[J ; S x J H . with x J [ TW J for J ; S. For I, J ; S and d g DI J , the A-linear map f IdJ on [J 9; S x J 9 H defined by
f IdJ : x J 9Tw ª d J , J 9TW I wW J Tw for all w g DJ 9 is an H-homomorphism. By Mackey decomposition, we obtain that the elements f IdJ < I, J ; S, d g DI J 4 form a basis of the algebra A. Let DŽW . be the set of all double cosets of W. We denote f IdJ by f D where D s WI dWJ . Let qw s q s i ??? q s i if si1 ??? si r is a reduced expres1 r sion of w. Since q s s qt if s and t are W-conjugate, qw is independent of the reduced expression of w. The following result can be proved by the same method used in wDu2, 1.4x. ŽIn wDR2x, we give a proof of this result.. For D s W1wWJ with w g DI J , let
Ž3.2. PROPOSITION.
u D s qw1r2 J
Ý D 9gD ŽW . zgD 9lD I J
a z , w fD 9 ,
where a z, w is defined by Bw D s Ý z g D 9l D I J , DŽW .4 is a basis of A. Ž3.3. DEFINITION.
w D 9 F w D a z, w TD 9.
Then u D < D g
For any x, y, z l , let f x, y, z g A be defined by Bx B y s
Ý
f x , y , z Bz .
zgW
For any D s WI xWJ , D 1 s WI9 yWJ 9 , and D 2 s WI0 zWJ 0 with x g Dq IJ , q y g Dq and z g D , let f s f . I9, J 9 I0 , J 0 D, D 1 , D 2 x, y, z Ž3.4. PROPOSITION. any J ; S. Then
Ž 3.5.
uD uD1 s
r2 Keep the setup abo¨ e. Let h J s qy1 Ý w g W J qw for wJ
½
hy1 J Ý fD , D1 , D 2 uD 2 ,
if J s I9,
0,
otherwise.
Proof. This follows from an argument similar to wDu1, 3.4x. From here onwards, we consider the Hecke algebra H associated to U2 nŽ q . or U2 nq1Ž q . over A s Zw t, ty1 x. Let A be the endomorphism algebra associated to H . First, we introduce the notion of cells in A following wLu2, Sect. 29.4x as follows. Let B s b [ u D < D g WI R WrWJ , I, J ; S4 be the Kazhdan] Lusztig basis of A defined as above. Let c b, b9, b0 g A with b, b9, b0 g B
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be defined by bb9 s Ý b0 g B c b, b9, b0 b0. Following wLu2, Sect. 29.4x, we denote by b9 FL b Žresp. b9 FR b . for b, b9 g B if there exists a sequence b1 s b, b 2 , . . . , bn s b9 and a 1 , . . . , a ny1 in B such that ca i , b i , b iq 1 / 0 Žresp. c b , a , b / 0. for all i g 1, 2, . . . , n y 14 . Let FL R be the preorder i i iq 1 on B generated by FL and FR . The corresponding equivalence relations are denoted by ;L , ;R , and ;L R . The corresponding equivalence classes are called left cells, right cells, and two-sided cells of B, respectively. By Ž3.5., we have the following proposition. Ž3.6. PROPOSITION. For any double coset D g DŽW ., let wD be the distinguished double coset representati¨ e of maximal length. For any u D , u D 9 g A, u D ;L R u D 9 if and only if wD ;L R wD 9. Ž3.7. PROPOSITION. For any u D g A and D g DI, J , there are two tableaux P g T0 Ž m , l. and Q g T0 Ž m , n . corresponding to wD where m is the twosided cell of S 2 n containing wD , and l, n are two compositions of 2 n determined by I and J, respecti¨ ely Ž e. g., if I s s14 , then c Ž s1 . s t 1 t 2 ny1 s w 0, l where l is the composition Ž2, 12 ny4 , 2. of 2 n.. Moreo¨ er, if u D ;L u D 9 then Q s Q9 and if u D ;R u D 9 then P s P9, where P9, Q9 are determined by u D 9. Proof. By assumption, wD g Dq I J . Let wD g m , where m is a two-sided cell of W Ž Bn .. There is a two-sided cell of S 2 n containing wD if we consider wD as an element of S 2 n . We denote this two-sided cell of S 2 n by m , too. Let Gm be the left cell of S 2 n containing w 0, m , the longest word in the standard parabolic subgroup of S 2 n with respect to m. Consider the elements x, y g S 2 n such that tŽ x . s tŽ wD ., sŽ x . s sŽ w 0, m ., tŽ y . s tŽ w 0, m ., and sŽ y . s sŽ wD .. By Ž2.5., wD ;L y ;R w 0, m and wD ;R x ;L w 0, m . Hence L Ž x . s L Ž wD .
and
RŽ x . s m ,
L Ž y. s m,
and
R Ž y . s R Ž wD . ,
where L Ž w . s s g S N sw - w4 and RŽ w . s s g S N ws - w4 . So, x x g Dlq, m and yqg Dnq, m where l and n are two composition of 2 n determined by c Ž wI . and c Ž wJ .. Since the Robinson]Schensted map gives a bijection between the elements in a symmetry group and the set of standard tableaux, x and y are determined uniquely by wD . BywDu3, 3.3x, there is a bijection between Dlq, m l Gm and the set of semistandard m-tableaux of l-type. x x Žresp. yq. determines a unique element P Žresp. Q . in T0 Ž m , l. Žresp. T0 Ž m , n ... Let x 1 , y 1 and x 2 , y 2 be the elements in S 2 n determined by wD and wD 9. If u D ;L u D 9 , then wD ;L wD 9 in W Ž Bn . by Ž3.5.. Hence wD ;L wD 9 in S 2 n by the remark after Ž1.5. if we consider wD , wD 9 as elements in S 2 n . By construction, y 1 s y 2 . Since Q1 and Q2 are uniquely determined by y 1 and y 2 , Q1 s Q2 . One can prove P1 s P2 if u D ;R u D 9 similarly.
ENDOMORPHISM ALGEBRAS
317
Note that P and Q are uniquely determined by wD . Hereafter, we denote such P and Q by P Ž wD . and QŽ wD ., respectively. Ž3.8. DEFINITION.
Let V be a two-sided cell of A. Let
I Ž V . s P P s P Ž wD . , for all u D g V 4 J Ž V . s Q Q s Q Ž wD . , for all u D g V 4 . Obviously, I Ž V . s J Ž V .. Now we prove the main result of this paper. Ž3.9. THEOREM. Let H be the generic Hecke algebra associated to U2 nŽ q . or U2 nq1Ž q . and let A be the endomorphism algebra associated to H . Let L be the set of two-sided cells of A. Then the Kazhdan]Lusztig basis for endomorphism algebra End H Ž[J ; S x J H . is a standard base on the poset L in the sense of Ž2.1.. Proof. We shall denote by L the set of all generalized two-sided cells of W Ž Bn . by Proposition 3.6. For any u D g A, let u D [ u P,m Q , where P s P Ž wD ., Q s QŽ wD ., and m is the two-sided cell containing wD Žsee Ž3.7... L is a poset with partial order F , which is the reverse order of FL R . For D g WI R WrWJ and D9 g WK R WrWL , let u D [ u Pl, Q and u D 9 [ u Pm1 , Q 1. By Ž3.5. and Ž3.6.,
u D u Pm1 , Q 1 '
Ý P 2gI Ž m .
f P 2 , m Ž u D , P1 . u Pm2 , Q 1 mod A ) m ,
where f P 2 , mŽ u D , P1 . [ hmy1 f D, D 9, D 0 s hmy1 f w D , w D 9, w D 0 for some double coset D0 g DŽW . Žsee Proposition 3.4.. We claim f P 2 , mŽ u D , P1 . is independent of Q1. In fact, consider the pairs Ž P1 , Q1 . and Ž P1 , QX1 . which are determined by wD 9 and wD 1. Let Gm be the left cell of S 2 n containing w 0, m , where w 0, m is the longest word in the standard parabolic subgroup of S 2 n with respect to m. By Proposition 3.7, if we denote by x the element in Gm corresponding to P1 , then wD 9 ;R x ;R wD 1. Since the Kazhdan]Lusztig basis of H is a standard base we have f w D , w D 9, w D 0 s f w D , w D , w D 0 for wD 9 ;R 1 wD 1 , and the claim follows. Similarly, we have the result for u P,l Q u D 9. Thus A is a standardly based algebra on the poset L with standard base u D . Ž3.10. Remark. The above proof is for the case G s U2 nŽ q .. One can prove the case G s U2 nq1Ž q . similarly. Let u, ¨ be two parameters. The Hecke algebras associated to Weyl groups of type Bn over the group ring Zw u, uy1 , ¨ , ¨ y1 x have been proved to be cellular algebras, a special case of standardly based algebras Žsee wGLx.. However, we have not found a total
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order on Zw u, uy1 , ¨ , ¨ y1 x such that the Kazhdan]Lusztig base Bw < w g W Ž Bn .4 is a standard base. If we prove that there is a total order on Zw u, uy1 ,¨ , ¨ y1 x such that the corresponding Kazhdan]Lusztig basis is a standard base, then one can prove that the endmorphism algebra A is a standardly based algebra. Ž3.11. COROLLARY. For a commutati¨ e noetherian ring R o¨ er A s Zw t, ty1 x with unit 1, let HR [ H mA R and let AR [ End HRŽ[j : s X J HR .. Then the Kazhdan]Lusztig basis AR is a standardly based algebra on the poset L, where L is the set of two-sided cells of A.
4. REPRESENTATION OF AK OVER A FIELD K Let K be a field. Assume there is a map from A s Zw t, ty1 x to K such that the image of t is not zero. Let AK s A mA K. In this section, we shall describe the simple modules and standard modules in the category of left AK-modules. First, we define a bilinear form Žsee wDR1, 1.2.6x.. Let L be the set of generalized two-sided cells of W Ž Bn .. For any l g L, let I Ž l. and u Pm, Q < P, Q g I Ž m ., m g L4 be defined as in Ž3.8. and Ž3.9.. Ž4.1. Bilinear Form. Keep the step above. For each l g L, let function fl: I Ž l. = I Ž l. ª k be defined as follows: For any u Pl, Q , u Pl9, Q9 , there is a unique element flŽ Q, P9. g K such that
u Pl, Q u Pl9, Q9 ' fl Ž Q, P9 . u Pl, Q9 mod Ž AK) l . . This is well defined by Ž3.9. Žcompare wDR1, 1.2.3x.. Ž4.2. Standard modules. ŽCompare wDR1, 2.1x.. For l g L, we define DŽ l. to be the left AK-module with K-basis u Pl < P g I Ž l.4 and module action defined by
Ž 4.3.
au Pl s
Ý P 9gI Ž l .
f P 9, l Ž a, P . u Pl9 .
We call DŽ l. the standard module of the category of left AK-modules. Ž4.4. THEOREM.
Let L 0 [ l g L < fl / 04 .
Ž1. If l g L 0 , then the standard module DŽ l. has a simple head LŽ l., which is absolutely irreducible. Ž2. LŽ l. < l g L 0 4 is a complete set of non-isomorphic simple AKmodules. Ž3. If LŽ l. Ž l g L 0 . is a composition factor of DŽ m ., then l F m. Ž4. The decomposition matrix of AK is upper unitriangular.
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319
Proof. Theorem Ž4.4. follows from the results on representation theory of standardly based algebras in wDR1x Žsee wloc. cit. 2.4.2, 2.4.6x.. ACKNOWLEDGMENTS I wish to thank the referee for many helpful comments. Part of this work was done while I visited the University of New South Wales. I wish to thank the University of New South Wales for its hospitality during my visit.
REFERENCES wBx wBrex
C. Berge, ‘‘Principles of Combinatorics,’’ Academic Press, New York, 1971. K. Bremke, ‘‘Kazhdan]Lusztig Polynomials and Cells for Affine Weyl Groups and Unequal Parameters,’’ Ph.D. dissertation, MIT, 1996. wBVx D. Barbasch and D. Vogan, Primitive ideals and orbital integrals in complex classical groups, Math. Ann. 259 Ž1982., 153]199. wCurx C. W. Curtis, On Lusztig’s isomorphism theorem for Hecke algebras, J. Algebra 92 Ž1985., 348]365. wCPS1x E. Cline, B. Parshall, and L. Scott, Finite dimensional algebras and highest weight categories, J. Reine, Angew. Math. 391 Ž1988., 85]99. wCPS2x E. Cline, B. Parshall, and L. Scott, Integral and graded quasi-hereditary algebras, I, J. Algebra 131 Ž1990., 126]160. wDJ1x R. Dipper and G. D. James, The q-Schur algebra, Proc. London Math. Soc. Ž 3 . 59 Ž1989., 23]50. wDJ2x R. Dipper and G. D. James, q-tensor space and q-Weyl modules, Trans. Amer. Math. Soc. 327 Ž1991., 251]282. wDu1x J. Du, Kazhdan]Lusztig bases and isomorphism theorems for q-Schur algebras, Contemp. Math. 139 Ž1991., 121]140. wDu2x Jie Du, IC-bases and Quantum Linear Groups, Proc. Sympos. Pure Math. 56 Ž1994., 135]148. wDu3x J. Du, Canonical bases for irreducible representations of quantum GLn , II, J. London Math. Soc. Ž 2 . 51 Ž1995., 461]470. wDR1x J. Du and H. Rui, Based algebras and standard bases for quasi-hereditary algebras, UNSW Preprint No. 8, 1996. wDR2x J. Du and H. Rui, Cells of finite Coxeter groups in multiparameter case, preprint, 1997. wGrax J. Graham, ‘‘Modular Representations of Hecke Algebras and Related Algebras,’’ Ph.D. dissertation, Sydney University, 1995. wGx J. Graham and G. Lehrer, Cellular algebras, In¨ ent. Math. 123 ŽŽ1996., 1]34. wKLx D. Kazhdan and G. Lusztig, Presentation of Coxeter groups and Hecke algebras, In¨ ent. Math. 53 Ž1979., 155]174. wLu1x G. Lusztig, ‘‘Left Cells in Weyl Groups,’’ Lecture Notes in Mathematics, Vol. 1024, pp. 99]111, Springer-Verlag, New YorkrBerlin, 1983. wLu2x G. Lusztig, Introduction to quantum groups, Prog. Math. 110 Ž1993.. wPWx B. Parshall and J. Wang, Quantum Linear Groups, Mem. Amer. Math. Soc. 89, No. 439, Ž1991.. wSx M. P. Schutzenberger, Quelques remarques sur une construction de Schensted, ¨ Math. Scand. 12 Ž1963., 117]128.
JOURNAL OF ALGEBRA ARTICLE NO.
On Endomorphism Algebras Arising from Multiparameter Hecke Algebras Hebing Rui*, † Department of Mathematics, East China Uni¨ ersity of Technology, Shanghai, 200093, People’s Republic of China Communicated by Gordon James Received November 7, 1996
INTRODUCTION Let GŽ q .4 be a family of finite groups of Lie type having ŽW, S . as its irreducible Coxeter system. For each parameter q, let c s be the integer satisfying < B Ž q .: sB Ž q . l B Ž q .< s q c s for each s g S, where B Ž q . is a fixed Borel subgroup of GŽ q .. Then c s s c t if s and t are W-conjugate. Let Zw q 1r2 , qy1r2 x be the ring of Laurent polynomials in indeterminate q 1r2 . Then the Hecke algebra H corresponding to GŽ q . over A has a standard basis Tw < w g W 4 satisfying the conditions
Ž Ts y q c . Ž Ts q 1 . s 0,
if s g S,
Tx Ty s Tx y ,
if l Ž xy . s l Ž x . q l Ž y . ,
s
where l Ž . is the length function on W. For any I ; S, let x I s Ý w g W I Tw , where WI is the standard parabolic subgroup of W generated by I. Let A s End H Ž[I ; S x I H .. If GŽ q . is the general linear group GLŽ n, q ., then A is known as a q-Schur algebra Žsee wDJ1, DJ2x.. In wPWx, q-Schur algebras have been proved to be quasihereditary algebras in the sense of wCPS1x. In other words, the category of left A-modules is a highest weight category in the sense of wloc. cit.x. * The author gratefully acknowledges support by the National Natural Science Foundation 19501016 in China and Large ARC Grant A69530243 from the Australian Research Council. † E-mail: [email protected]. 308 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
ENDOMORPHISM ALGEBRAS
309
In this paper, we consider the cases when G is either U2 nŽ q . or U2 nq1Ž q ., where UnŽ q . is the group of unitary n by n matrices over the field of q elements. The Weyl group W associated to G is of type Bn . The Dynkin diagram of W is as follows: 1
`
2 n } ` } ??? s ` .
Let c s be the integers associated to G. If G s U2 nŽ q ., then cs s
½
1, 2,
if s s sn , otherwise.
cs s
½
3, 2,
if s s sn , otherwise.
If G s U2 nq1Ž q ., then
Let H be the Hecke algebra corresponding to either U2 nŽ q . or U2 nq1Ž q .. Let A be the endomorphism algebra associated to H . In this paper, we study the representation theory of endomorphism algebra A by using Kazhdan]Lusztig theory. Our method is based on the representation theory of standardly based algebras given in wDR1x. The main result of this paper is that the Kazhdan]Lusztig basis of A is a standard base in the sense of wloc. cit., 1.2.1x Žsee Ž2.1... Using the results in wloc. cit.x, we describe the simple modules and standard modules in the category of left A-modules. Our result shows that the category of left A-modules has results similar to the case of a highest weight category Žsee Ž4.4... When we specialize q to 1, we get the results on endomorphism algebras End kW Ž[J : S TJ . where W is the Weyl group of type Bn and TJ is the permutation module on the cosets WJ w4w g W . The contents of this paper are organized as follows. In Section 1, we recall some basic definitions and results. Let H be the Hecke algebra associated to U2 nŽ q . or U2 nq1Ž q .. In Section 2, we prove that the Kazhdan]Lusztig basis of H is a standard base by a result due to Graham in wGrax. In Section 3, we prove the Kazhdan]Lusztig basis of A is a standard base. In Section 4, we give some results on the representation theory of endomorphism algebra A.
1. HECKE ALGEBRAS Let W be a finite Coxeter group with S as its distinguished generator set of simple reflections. Let A s Zw t, ty1 x be the ring of Laurent polyno-
310
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mials in indeterminate t. For any s g S, let c s be a positive integer such that c s s c t if s and t are W-conjugate. Then the generic Hecke algebra H associated to W is an associative algebra over A with free A-basis Tw < w g W 4 . Let l Ž . be the length function on W. Then,
Ž Ts y t 2 c . Ž Ts q 1. s 0,
if s g S,
Tx Ty s Tx y ,
if l Ž xy . s l Ž x . q l Ž y . .
s
Ž 1.1.
If the parameters c s are determined by finite groups GŽ q . and W the Weyl group associated to GŽ q ., we say the generic Hecke algebra H is associated to GŽ q .. Ž1.2. Let F be the Bruhat]Chevalley order on W. In wLu1x, Lusztig introduced another basis Bw < w g W 4 for H such that Bw s
Ý
PyU, w Ty ,
yFw
where Py,U w is known as the Kazhdan]Lusztig polynomial Žresp. generalized Kazhdan]Lusztig polynomial. if H has equal parameters Žresp. unequal parameters.. Ž1.3. Cells. Let FL be the preorder on W such that x FL y is there is z g W such that B x appears in the expression of Bz B y with non-zero coefficient. Let FR be the preorder on W such that x FR y if xy1 FL yy1 . Let FL R be the preorder on W generated by FL and FR. The equivalent relations corresponding to x g L, R, LR4 are denoted by ;x and the equivalent classes corresponding to L, R, LR are called left cells, right cells, and two-sided cells Žresp. generalized left, right and two-sided cells if H has unequal parameters.. Ž1.4. Let H be the generic Hecke algebra associated to finite groups U2 nŽ q . or U2 nq1Ž q .. Then H is of type Bn Žor Cn . with unequal parameters Žsee Introduction.. H is isomorphic to a subalgebra of a Hecke algebra associated to a certain symmetry group. In fact, if the Hecke algebra H is associated to U2 nŽ q ., then we consider the Hecke algebra associated to the symmetry groups S 2 n . For 1 F i F 2 n y 1, let t i s Ž i, i q 1. be the basic transposition. Then T s t i < 1 F i F 2 n y 14 is the distinguished generator set of S 2 n . Let a be the non-trivial isomorphism of S 2 n such that a ŽT . s T. Then the subgroup S 2an s w g S 2 n < a Ž w . s w4 is isomorphic to the Weyl group of type Bn with U s u i < 1 F i F n, u i s t i t 2 nyi if 1 F i F n y 1 and u n s t n 4 as its distinguished generator set of simple reflections. Let c i [ c u i s l Ž u i . where l Ž . is the length function on S 2 n . Let H Ž S 2an . be the Hecke
311
ENDOMORPHISM ALGEBRAS
algebra associated to S 2an with respect to c i . Then H Ž S 2an . ( H ,
Ž 1.5.
where H is the Hecke algebra associated to U2 nŽ q .. If H is associated to the finite group U2 nq1Ž q ., then H is isomorphic to a subalgebra of the Hecke algebra associated to the symmetry group S 2 nq1. Hereafter, we deal with the Hecke algebra with parameters associated to U2 nŽ q .. The second case can be dealt with similarly. Let W Ž Bn . be the Weyl group of type Bn with S s s1 , s2 , . . . , sn4 as its distinguished generator set of simple reflections. Let c : W Ž Bn . ª S 2an be such that c Ž si . s u i for 1 F i F n. Then c is an isomorphism. By Ž1.5., we have, for any x, y g W Ž Bn ., x ;L y if and only if c Ž x . ;L c Ž y . in S 2an . The last condition implies c Ž x . ;L c Ž y . in S 2 n by the result due to Lusztig in wLu1, Sect. 9.2x Žsee also wBre, Sect. 4.3x.. From here onwards, we consider w g W Ž Bn . as w g S 2an by abusing notation. 2. STANDARDLY BASED ALGEBRAS In this section, we prove the Kazhdan]Lusztig basis for Hecke algebras associated to U2 nŽ q . or U2 nq1Ž q . is a standard base introduced in wDR1x. First, we recall the definition of standard base. Ž2.1. Assume that R is a commutative ring with unit 1. Let A be an R-algebra and Ž L, F. a poset. A is called a standardly based algebra on L if the following conditions hold. Ža. For any l g L, there are index sets I Ž l. and J Ž l. such that B l s a il, j Ž i , j . g I Ž l . = J Ž l . .
½
5
Žb. The disjoint union B s Dlg L B l is a free R-basis of A. Žc. For any a g A, a i,l j g B, we have
Ž 2.2.
a ? a il, j '
Ý i9gI Ž l .
a il, j ? a '
Ý j9gJ Ž l .
l )l f i9, l Ž a, i . a i9, . j mod Ž A
fl , j9 Ž j, a . a il, j9 modŽ A ) l . ,
where A ) l is the R-submodule of A spanned by B m with m ) l and f i9, lŽ a, i ., fl, j9Ž j, a. g R are independent of j and i, respectively. Such a base B is said to be a standard base for the standardly based algebra A.
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HEBING RUI
Standardly based algebras are generalized versions of cellular algebras introduced by Graham and Lehrer in wGLx. Ž2.3. A composition l of a non-negative integer n is a sequence Ž l1 , l 2 , . . . , . such that Ý`is1 l i s n. If l i G l j for all i F j, then l is said to be a partition. For positive integer n, r, let LŽ n, r . Žresp. LqŽ n, r .. be the set of all compositions Žresp. partitions. of r with at most n parts. For l g LqŽ n, r ., we may identify l with its corresponding diagram which consists of crosses arranged in a manner as illustrated by the example l s Ž421. g LqŽ 3, 7. for which we have = ls = =
= =
=
=
A l-tableau t is obtained by replacing each cross by one of the numbers 1, 2, . . . , r. The symmetric group W s S r acts on the set of l-tableaux by letter permutations. For any partition l of r, we consider l-tableaux such that the number of entries i is equal to m i , where m is a composition of r. Such tableaux are called l-tableaux of type m. Let T Ž l, m . denote the set of all l-tableaux of type m. A l-tableau t is called semistandard Žresp. standard. if its entries are weakly increasing Žresp. strictly increasing . along each row and increasing along each column. Let T0 Ž l, m . be the set of all semistandard l-tableaux of type m. If n G r, we set v s Ž1r .. Clearly, T0 Ž l, v . is the set of all standard l-tableaux. Ž2.4. Robinson]Schensted Map. Let W be the symmetry group S r . Let f : w ¬ ŽtŽ w ., sŽ w .. be the Robinson]Schensted map from W to the set of pairs of standard tableaux Žsee, eg., wBx.. In wBVx, Barbasch and Vogan proved that, for x, y g W,
Ž 2.5.
x ;L y Ž resp. x ;R y . if and only if s Ž x . s s Ž y .
Ž resp. t Ž x . s t Ž y . . . Ž2.6. Let a be the non-trivial automorphism on S 2 n preserving the generator set T Žsee Ž1.4... In wLu1x and wBre, Sect. 4.3x, Lusztig and Bremke proved that any generalized left cell G of S 2an satisfies G s G1a for some left cell G1 of S 2 n . Let L be the set of generalized two-sided cells of W Ž Bn .. For any V g L, let I Ž V . s t g T0 Ž l , v . t s t Ž w . for some w g V 4 J Ž V . s s g T0 Ž l , v . s s s Ž w . for some w g V 4 , where l is the two-sided cell of S 2 n containing V. ŽHere, we consider V as a two-sided cell of S 2an .. By the result due to Bremke and Lusztig Žsee
ENDOMORPHISM ALGEBRAS
313
wLu1, Theorem 11x or wBre, P. 63, line 3x., l a is a union of generalized two-sided cells of S 2an . In fact, this result can be deduced from wloc. cit. 4.3.8x.. Obviously, I Ž V . s J Ž V . for a generalized two-sided cell of W Ž Bn . since tŽ w . s sŽ wy1 . and sŽ w . s tŽ wy1 . for any w g S 2 n . I Ž V . is a subset of T0a Ž l, v .; the latter consists of the elements in T0 Ž l, v . which are fixed by an evacuation map defined by Schutzenberger in wSx. Such elements are ¨ called symmetric standard tableaux Žsee wGra, Chap. 2x.. The following result follows from a result due to Graham wloc. cit. Chap. 3x. Ž2.7. THEOREM. Let H be the Hecke algebra associated to finite groups U2 nŽ q . or U2 nq1Ž q .. Then the Kazhdan]Lusztig basis of H is a standard base in the sense of Ž2.1.. Proof. For any w g W Ž Bn ., let Bw [ BPl, Q where l is the generalized two-sided cell of W Ž Bn . containing w and Ž P, Q . s f Ž w . g I Ž l. = I Ž l.. Let L be the set of all generalized two-sided cells of W Ž Bn .. Then L is a poset with partial order F , where F is the reversed order of FL R . For any l g L, let H ) l be the free A-submodule of H spanned by BPm, Q with P, Q g I Ž m . and m ) l. For all h g H Ž Bn ., we have
Ž 2.8.
hBPl, Q '
Ý f P 9l Ž h, P . BPl9, Q mod H ) l
BPl, Q h '
Ý fl , Q9 Ž Q, h . BPl, Q9 mod H ) l
by Ž2.5. and Corollary Ž4.3.10. in wBrex Žthe latter implies that if z9 FL z and z ;L R z9, then z ;L z9.. Moreover, f P 9 lŽ h, P . and fl , Q9Ž Q, l. are elements in A, which are independent of Q and P, respectively, by Ž1.5. and wGra, 3.2x Žthe latter says that the left cell modules EŽ G1 . and EŽ G2 . are isomorphic if G1 and G2 are in the same generalized two-sided cell of H Ž S 2an ... Thus the Kazhdan]Lusztig basis of H is a standard base in the sense of Ž2.1.. Let R be a commutative noetherian ring over A with unit 1. Let q 1r2 g R be the image of t. Let HR s H mA R. Then HR is the Hecke algebra over the commutative noetherian ring R. We denote Tw and Bw by Tw m 1 and Bw m 1 by abusing notation. By Theorem Ž2.7., we have the following result. Ž2.9. COROLLARY. Let HR be the Hecke algebra associated to one of the finite groups U2 nŽ q . or U2 nq1Ž q .. Then the Kazhdan]Lusztig basis Bw < w g W Ž Bn .4 is a standard base of HR in the sense of Ž2.1..
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HEBING RUI
3. ENDOMORPHISM ALGEBRAS End H Ž[J : S x J H . In this section, we prove our main result, which says the KzahdanLusztig basis of End H Ž[J : S x J H . is a standard base if H is the Hecke algebra associated to U2 nŽ q . or U2 nq1Ž q .. At first, we introduce the Kazhdan]Lusztig basis for End H Ž[J : S x J H .. When H has equal parameters, the Kazhdan]Lusztig basis for End H Ž[J : S x J H . has been introduced by Du in wDu1x. Let W be a finite Coxeter group of non-simply laced type. Let S be its distinguished generator set of simple reflections. For each subset I ; S, let WI be the parabolic subgroup of W generated by I, and DI the set of representatives of minimal length in the right cosets WI R W. Thus, the set DI J s DI l Dy1 for I, J ; S consists of the representatives of minimal J length in the double cosets WI R WrWJ . Let Dq I J be the set of the representatives of maximal length in the double cosets WI R WrWJ . Let G be an abelian group. We now assume that a total order e on G is given, which is compatible with the structure of G. For any s g S, we associate a parameter q s g G such that q s s qt if s, t are W-conjugate. We also assume that q s g Gq for all s g S, where Gq is the set of elements which are strictly positive for the total order e. Let H be the Hecke algebra associated to W over Zw G x. Then H has a free Zw G x-basis Tw < w g W 4 such that
Ž Ts y qs . Ž Ts q 1 . s 0,
if s g S,
Tx Ty s Tx y ,
if l Ž xy . s l Ž x . q l Ž y . .
If G s ² t 1r2 :, and q s s t 2 c s for some integer c s satisfying c s s c t if s and t are W-conjugate, then the Hecke algebra defined above coincides with that defined in Ž1.1.. Let Bwe < w g W 4 be the Kazhdan]Lusztig basis with respect to the total order e Žsee wLu1x.. Hereafter, we use Bw to replace Bwe by abusing notation. For any D ; W, let TD s Ý x g D Tx . For any I, J : S, define HI J to be the A-free module spanned by the elements TD , D g WI R WrWJ . A direct computation shows that HI J can be characterized as follows: HI J s h g H < Ts h s qs h, hTt s qt h, for all s g I and t g J 4 . Ž3.1. PROPOSITION ŽCompare wCur, 1.10x.. Let H be the Hecke algebra o¨ er Zw G x, where G is an abelian group. Let e be a total order on G. Then Kazhdan]Lusztig base elements Bw 4w g D qI J with respect to e are a free Zw G x-basis for HI J .
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ENDOMORPHISM ALGEBRAS
Proof. This follows from an argument similar to wCur, 1.10x. Now we introduce the Kazhdan ] Lusztig basis for A [ End H Ž[J ; S x J H . with x J [ TW J for J ; S. For I, J ; S and d g DI J , the A-linear map f IdJ on [J 9; S x J 9 H defined by
f IdJ : x J 9Tw ª d J , J 9TW I wW J Tw for all w g DJ 9 is an H-homomorphism. By Mackey decomposition, we obtain that the elements f IdJ < I, J ; S, d g DI J 4 form a basis of the algebra A. Let DŽW . be the set of all double cosets of W. We denote f IdJ by f D where D s WI dWJ . Let qw s q s i ??? q s i if si1 ??? si r is a reduced expres1 r sion of w. Since q s s qt if s and t are W-conjugate, qw is independent of the reduced expression of w. The following result can be proved by the same method used in wDu2, 1.4x. ŽIn wDR2x, we give a proof of this result.. For D s W1wWJ with w g DI J , let
Ž3.2. PROPOSITION.
u D s qw1r2 J
Ý D 9gD ŽW . zgD 9lD I J
a z , w fD 9 ,
where a z, w is defined by Bw D s Ý z g D 9l D I J , DŽW .4 is a basis of A. Ž3.3. DEFINITION.
w D 9 F w D a z, w TD 9.
Then u D < D g
For any x, y, z l , let f x, y, z g A be defined by Bx B y s
Ý
f x , y , z Bz .
zgW
For any D s WI xWJ , D 1 s WI9 yWJ 9 , and D 2 s WI0 zWJ 0 with x g Dq IJ , q y g Dq and z g D , let f s f . I9, J 9 I0 , J 0 D, D 1 , D 2 x, y, z Ž3.4. PROPOSITION. any J ; S. Then
Ž 3.5.
uD uD1 s
r2 Keep the setup abo¨ e. Let h J s qy1 Ý w g W J qw for wJ
½
hy1 J Ý fD , D1 , D 2 uD 2 ,
if J s I9,
0,
otherwise.
Proof. This follows from an argument similar to wDu1, 3.4x. From here onwards, we consider the Hecke algebra H associated to U2 nŽ q . or U2 nq1Ž q . over A s Zw t, ty1 x. Let A be the endomorphism algebra associated to H . First, we introduce the notion of cells in A following wLu2, Sect. 29.4x as follows. Let B s b [ u D < D g WI R WrWJ , I, J ; S4 be the Kazhdan] Lusztig basis of A defined as above. Let c b, b9, b0 g A with b, b9, b0 g B
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be defined by bb9 s Ý b0 g B c b, b9, b0 b0. Following wLu2, Sect. 29.4x, we denote by b9 FL b Žresp. b9 FR b . for b, b9 g B if there exists a sequence b1 s b, b 2 , . . . , bn s b9 and a 1 , . . . , a ny1 in B such that ca i , b i , b iq 1 / 0 Žresp. c b , a , b / 0. for all i g 1, 2, . . . , n y 14 . Let FL R be the preorder i i iq 1 on B generated by FL and FR . The corresponding equivalence relations are denoted by ;L , ;R , and ;L R . The corresponding equivalence classes are called left cells, right cells, and two-sided cells of B, respectively. By Ž3.5., we have the following proposition. Ž3.6. PROPOSITION. For any double coset D g DŽW ., let wD be the distinguished double coset representati¨ e of maximal length. For any u D , u D 9 g A, u D ;L R u D 9 if and only if wD ;L R wD 9. Ž3.7. PROPOSITION. For any u D g A and D g DI, J , there are two tableaux P g T0 Ž m , l. and Q g T0 Ž m , n . corresponding to wD where m is the twosided cell of S 2 n containing wD , and l, n are two compositions of 2 n determined by I and J, respecti¨ ely Ž e. g., if I s s14 , then c Ž s1 . s t 1 t 2 ny1 s w 0, l where l is the composition Ž2, 12 ny4 , 2. of 2 n.. Moreo¨ er, if u D ;L u D 9 then Q s Q9 and if u D ;R u D 9 then P s P9, where P9, Q9 are determined by u D 9. Proof. By assumption, wD g Dq I J . Let wD g m , where m is a two-sided cell of W Ž Bn .. There is a two-sided cell of S 2 n containing wD if we consider wD as an element of S 2 n . We denote this two-sided cell of S 2 n by m , too. Let Gm be the left cell of S 2 n containing w 0, m , the longest word in the standard parabolic subgroup of S 2 n with respect to m. Consider the elements x, y g S 2 n such that tŽ x . s tŽ wD ., sŽ x . s sŽ w 0, m ., tŽ y . s tŽ w 0, m ., and sŽ y . s sŽ wD .. By Ž2.5., wD ;L y ;R w 0, m and wD ;R x ;L w 0, m . Hence L Ž x . s L Ž wD .
and
RŽ x . s m ,
L Ž y. s m,
and
R Ž y . s R Ž wD . ,
where L Ž w . s s g S N sw - w4 and RŽ w . s s g S N ws - w4 . So, x x g Dlq, m and yqg Dnq, m where l and n are two composition of 2 n determined by c Ž wI . and c Ž wJ .. Since the Robinson]Schensted map gives a bijection between the elements in a symmetry group and the set of standard tableaux, x and y are determined uniquely by wD . BywDu3, 3.3x, there is a bijection between Dlq, m l Gm and the set of semistandard m-tableaux of l-type. x x Žresp. yq. determines a unique element P Žresp. Q . in T0 Ž m , l. Žresp. T0 Ž m , n ... Let x 1 , y 1 and x 2 , y 2 be the elements in S 2 n determined by wD and wD 9. If u D ;L u D 9 , then wD ;L wD 9 in W Ž Bn . by Ž3.5.. Hence wD ;L wD 9 in S 2 n by the remark after Ž1.5. if we consider wD , wD 9 as elements in S 2 n . By construction, y 1 s y 2 . Since Q1 and Q2 are uniquely determined by y 1 and y 2 , Q1 s Q2 . One can prove P1 s P2 if u D ;R u D 9 similarly.
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Note that P and Q are uniquely determined by wD . Hereafter, we denote such P and Q by P Ž wD . and QŽ wD ., respectively. Ž3.8. DEFINITION.
Let V be a two-sided cell of A. Let
I Ž V . s P P s P Ž wD . , for all u D g V 4 J Ž V . s Q Q s Q Ž wD . , for all u D g V 4 . Obviously, I Ž V . s J Ž V .. Now we prove the main result of this paper. Ž3.9. THEOREM. Let H be the generic Hecke algebra associated to U2 nŽ q . or U2 nq1Ž q . and let A be the endomorphism algebra associated to H . Let L be the set of two-sided cells of A. Then the Kazhdan]Lusztig basis for endomorphism algebra End H Ž[J ; S x J H . is a standard base on the poset L in the sense of Ž2.1.. Proof. We shall denote by L the set of all generalized two-sided cells of W Ž Bn . by Proposition 3.6. For any u D g A, let u D [ u P,m Q , where P s P Ž wD ., Q s QŽ wD ., and m is the two-sided cell containing wD Žsee Ž3.7... L is a poset with partial order F , which is the reverse order of FL R . For D g WI R WrWJ and D9 g WK R WrWL , let u D [ u Pl, Q and u D 9 [ u Pm1 , Q 1. By Ž3.5. and Ž3.6.,
u D u Pm1 , Q 1 '
Ý P 2gI Ž m .
f P 2 , m Ž u D , P1 . u Pm2 , Q 1 mod A ) m ,
where f P 2 , mŽ u D , P1 . [ hmy1 f D, D 9, D 0 s hmy1 f w D , w D 9, w D 0 for some double coset D0 g DŽW . Žsee Proposition 3.4.. We claim f P 2 , mŽ u D , P1 . is independent of Q1. In fact, consider the pairs Ž P1 , Q1 . and Ž P1 , QX1 . which are determined by wD 9 and wD 1. Let Gm be the left cell of S 2 n containing w 0, m , where w 0, m is the longest word in the standard parabolic subgroup of S 2 n with respect to m. By Proposition 3.7, if we denote by x the element in Gm corresponding to P1 , then wD 9 ;R x ;R wD 1. Since the Kazhdan]Lusztig basis of H is a standard base we have f w D , w D 9, w D 0 s f w D , w D , w D 0 for wD 9 ;R 1 wD 1 , and the claim follows. Similarly, we have the result for u P,l Q u D 9. Thus A is a standardly based algebra on the poset L with standard base u D . Ž3.10. Remark. The above proof is for the case G s U2 nŽ q .. One can prove the case G s U2 nq1Ž q . similarly. Let u, ¨ be two parameters. The Hecke algebras associated to Weyl groups of type Bn over the group ring Zw u, uy1 , ¨ , ¨ y1 x have been proved to be cellular algebras, a special case of standardly based algebras Žsee wGLx.. However, we have not found a total
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order on Zw u, uy1 , ¨ , ¨ y1 x such that the Kazhdan]Lusztig base Bw < w g W Ž Bn .4 is a standard base. If we prove that there is a total order on Zw u, uy1 ,¨ , ¨ y1 x such that the corresponding Kazhdan]Lusztig basis is a standard base, then one can prove that the endmorphism algebra A is a standardly based algebra. Ž3.11. COROLLARY. For a commutati¨ e noetherian ring R o¨ er A s Zw t, ty1 x with unit 1, let HR [ H mA R and let AR [ End HRŽ[j : s X J HR .. Then the Kazhdan]Lusztig basis AR is a standardly based algebra on the poset L, where L is the set of two-sided cells of A.
4. REPRESENTATION OF AK OVER A FIELD K Let K be a field. Assume there is a map from A s Zw t, ty1 x to K such that the image of t is not zero. Let AK s A mA K. In this section, we shall describe the simple modules and standard modules in the category of left AK-modules. First, we define a bilinear form Žsee wDR1, 1.2.6x.. Let L be the set of generalized two-sided cells of W Ž Bn .. For any l g L, let I Ž l. and u Pm, Q < P, Q g I Ž m ., m g L4 be defined as in Ž3.8. and Ž3.9.. Ž4.1. Bilinear Form. Keep the step above. For each l g L, let function fl: I Ž l. = I Ž l. ª k be defined as follows: For any u Pl, Q , u Pl9, Q9 , there is a unique element flŽ Q, P9. g K such that
u Pl, Q u Pl9, Q9 ' fl Ž Q, P9 . u Pl, Q9 mod Ž AK) l . . This is well defined by Ž3.9. Žcompare wDR1, 1.2.3x.. Ž4.2. Standard modules. ŽCompare wDR1, 2.1x.. For l g L, we define DŽ l. to be the left AK-module with K-basis u Pl < P g I Ž l.4 and module action defined by
Ž 4.3.
au Pl s
Ý P 9gI Ž l .
f P 9, l Ž a, P . u Pl9 .
We call DŽ l. the standard module of the category of left AK-modules. Ž4.4. THEOREM.
Let L 0 [ l g L < fl / 04 .
Ž1. If l g L 0 , then the standard module DŽ l. has a simple head LŽ l., which is absolutely irreducible. Ž2. LŽ l. < l g L 0 4 is a complete set of non-isomorphic simple AKmodules. Ž3. If LŽ l. Ž l g L 0 . is a composition factor of DŽ m ., then l F m. Ž4. The decomposition matrix of AK is upper unitriangular.
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319
Proof. Theorem Ž4.4. follows from the results on representation theory of standardly based algebras in wDR1x Žsee wloc. cit. 2.4.2, 2.4.6x.. ACKNOWLEDGMENTS I wish to thank the referee for many helpful comments. Part of this work was done while I visited the University of New South Wales. I wish to thank the University of New South Wales for its hospitality during my visit.
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