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A Frobenius Formula for the Characters of Ariki–Koike Algebras
Journal of Algebra 226, 818᎐856 Ž2000. doi:10.1006rjabr.1999.8178, available online at http:rrwww.idealibrary.com on
A Frobenius Formula for the Characters of Ariki᎐Koike Algebras1 Toshiaki Shoji Department of Mathematics, Science Uni¨ ersity of Tokyo, Noda, Chiba 278-8510, Japan Communicated by Michel Broue´ Received June 10, 1999
Let Hn, r be the Ariki᎐Koike algebra associated to the complex reflection group Wn, r s GŽ r, 1, n.. In this paper, we give a new presentation of Hn, r by making use of the Schur᎐Weyl reciprocity for Hn, r established by M. Sakamoto and T. Shoji Ž1999, J. Algebra, 221, 293᎐314.. This allows us to construct various non-parabolic subalgebras of Hn, r . We construct all the irreducible representations of Hn, r as induced modules from such subalgebras. We show the existence of a partition of unity in Hn, r , which is specialized to a partition of unity in the group algebra ⺓Wn, r . Then we prove a Frobenius formula for the characters of Hn, r , which is an analogy of the Frobenius formula proved by A. Ram Ž1991, In¨ ent. Math. 106, 461᎐488. for the Iwahori᎐Hecke algebra of type A. 䊚 2000 Academic Press
Contents 1. Introduction 2. Cyclotomic Schur᎐Weyl reciprocity 3. A New Presentation of HK 4. Irreducible Representations of HK 5. Orthogonal Primiti¨ e Idempotents 6. A Frobenius Formula for the Characters of HK 7. Character Values
1. INTRODUCTION In wRx, Ram gave a Frobenius type formula for the characters of the Hecke algebra Hn associated to the symmetric group ᑭ n . He derived 1
This paper is a contribution to the Joint Research Project ‘‘Representation Theory of Finite and Algebraic Groups’’ 1997᎐1999 under the Japanese᎐German Cooperative Science Promotion Program supported by JSPS and DFG. 818 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
ARIKI ᎐ KOIKE ALGEBRAS
819
the formula by making use of the Schur᎐Weyl reciprocity between Hn and the quantum group Uq Ž gl m ., established by Jimbo wJx. Let Hn, r be the Ariki᎐Koike algebra, i.e., the cyclotomic Hecke algebra associated to the complex reflection group Wn, r s GŽ r, 1, n.. The Schur᎐Weyl reciprocity was generalized in wSSx to the case between Hn, r and a certain Levi subalgebra of Uq Ž gl m .. In this paper, we give a Frobenius type formula for the characters of Hn, r , based on the Schur᎐Weyl reciprocity mentioned above. ŽNote that in this introduction, the algebra Hn, r stands for HK in Section 2.. One of the crucial ingredients in wRx is the existence of a partition of unity in Hn , i.e., a complete system of primitive orthogonal idempotents, which is specialized to a partition of unity in the group algebra ⺓ ᑭ n . This result had been proved independently by Gyoja wGx and Wenzl wWx. We show that a similar fact also holds for Hn, r . ŽIn the first draft of the paper, this result was used to obtain the Frobenius formula for HK . However, as pointed out by the referee, there exists an alternative argument for it without relying on this fact. Nevertheless, we left the part discussing a partition of unity unchanged, as we believe that it is of independent interest.. First we give a new presentation of Hn, r based on the result in wSSx. We show that Hn, r is generated by 1 , . . . , n and a2 , . . . , a n , where a2 , . . . , a n are generators of Hn as usual, and 1 , . . . , n satisfy a common relation of degree r. Moreover, 1 , . . . , n generate an abelian subalgebra of Hn, r . Note that our abelian subalgebra is different from the one defined in wAKx. The relationship between our generators and generators a1 , . . . , a n given in wAKx is complicated. But since our generators j are symmetric for j s 1, . . . , n, it makes it possible to embed the algebra Hn1 , r m Hn 2 , r m ⭈⭈⭈ into Hn, r , whenever Ýn i F n, just as in the group case. ŽPfeiffer wPx considered certain non-parabolic subalgebras in the case of Iwahori᎐Hecke algebras of type B based on the standard generators. However, its structure is more complicated than ours.. Then we can construct all the irreducible representations of Hn, r as induced modules from certain Žnon-parabolic. subalgebras, in analogy to the group case construction. Under this setting, one can generalize the method of Gyoja in the case of type A to our case. A Frobenius formula is given as follows. We define some elements a in Hn, r , parametrized by multipartitions s Ž Ž1., . . . , Ž r . . of size n. Note that they are mapped, by the specialization to ⺓Wn, r , to a complete system of representatives of conjugacy classes in Wn, r . Let q be an irreducible character of Hn, r associated to a multipartition of size n. Then we have q˜ Ž x ; q, u . s
Ý q Ž a . S Ž x . ,
820
TOSHIAKI SHOJI
where q˜Ž x; q, u. is a certain polynomial, which can be described by using Hall᎐Littlewood symmetric functions qnŽ x; t ., and S Ž x . is the Schur function associated to a multipartition Žsee Section 6 for the detailed definition.. As an application one can show, for any character q of Hn, r , that the character values q Ž a w . are determined by a simple algorithm from the values q Ž a ., where a w ¬ w g Wn, r 4 is a basis of Hn, r defined in terms of our generators. The author is grateful to A. Ram for stimulating discussions on the occasion of a symposium in Kyoto, 1998. He also thanks the referee for his suggestions on a partition of unity of HK .
2. CYCLOTOMIC SCHUR᎐WEYL RECIPROCITY 2.1. Let R s ⺪w q, qy1 , u1 , . . . , u r x be a polynomial ring over ⺪ with indeterminates q, u1 , . . . , u r . The Ariki᎐Koike algebra, i.e., the cyclotomic Hecke algebra associated to the complex reflection group Wn, r s GŽ r, 1, n., is the algebra H s Hn, r over R with generators a1 , . . . , a n subject to the following conditions:
Ž a1 y u1 . ⭈⭈⭈ Ž a1 y u r . s 0, Ž ai y q . Ž ai q qy1 . s 0
Ž i G 2. ,
a1 a2 a1 a2 s a2 a1 a2 a1 , ai a j s a j ai a i a iq1 a i s a iq1 a i a iq1
Ž < i y j < G 2. , Ž 2 F i F n y 1. .
Let K be the quotient field of R. We set W s Wn, r . It is known by wAKx that H is a free R-module with rank < W < s n!r n, and that HK s K mR H is a semisimple algebra over K. r 2.2. Let V s [is1 Vi be a free R-module with Vi s R m i . We set r m s Ý is1 m i . Let us fix a basis E s ¨ 1 , . . . , ¨ m 4 of V such that
Ei s ¨ j ¬ m1 q ⭈⭈⭈ qm iy1 - j F m1 q ⭈⭈⭈ qm i 4 gives rise to a basis of Vi . We define a map b: 1, 2, . . . , m4 ª ⺞ by bŽ j . s i whenever ¨ j g Vi . Let wt: V ª V be a linear operator defined by wt Ž ¨ j . s u bŽ j.¨ j . Let us define linear operators, T, , S on V m2 as follows.
ARIKI ᎐ KOIKE ALGEBRAS
821
For ¨ i , ¨ j g E ,
¡q¨ m ¨ i
~
if i s j,
j
T Ž¨i m ¨j . s ¨ j m ¨i
¢¨ m ¨ q Ž q y q j
if i ) j,
y1
i
. ¨i m ¨j
if i - j,
Ž¨i m ¨j . s ¨j m ¨i , SŽ¨i m ¨j . s
½
T Ž¨i m ¨j .
if b Ž i . s b Ž j . ,
Ž¨i m ¨j .
if b Ž i . / b Ž j . .
Using these operators, we define operators Ti , i , Si , j g End R V mn, Ž2 F i F n, 1 F j F n., by the following rule: iy2. nyi. Ti s idmŽ m T m idmŽ , V V iy2. nyi. i s idmŽ m m idmŽ , V V iy2. nyi. Si s idmŽ m S m idmŽ , V ¨ jy1. nyj. j s idmŽ m wt m idmŽ . V V
We now define an operator T1 on V mn by T1 s Ty1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S2 1 . 2
Ž 2.2.1.
Then it is shown in wSS, Theorem 3.2x that : a i ¬ Ti Ž1 F i F n. gives rise to a representation of H on V mn. 2.3. Let Uq Ž gl m . be the quantized universal enveloping algebra of gl m over K. Hence, Uq Ž gl m . is an associative algebra with generators, e i , f i , Ž1 F i F m y 1. and q " i , Ž1 F i F m., and well-known relations Žsee, e.g., wSSx.. We consider the natural representation of Uq Ž gl m . on VK s K mR V, which is given by
Ž ei . ¨ j s Ž fi . ¨ j s Ž q" i .¨j s
½ ½
½
¨ jy1
if j s i q 1,
0
if j / i q 1,
¨ jq1
if j s i ,
0
if j / i ,
q"1¨ j
if j s i ,
0
if j / i.
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TOSHIAKI SHOJI
By the comultiplication ⌬: Uq Ž gl m . ª Uq Ž gl m . m Uq Ž gl m ., induces a representation of Uq Ž gl m . on VKmn, which we denote also by . It is expressed on generators as ny1
Ž ei . s
Ý
K im p m e i m 1mŽ ny1yp. ,
ps0 ny1
Ž fi . s
Ý
1m p m f i m Ž Ky1 i .
m Ž ny1yp .
,
ps0
Ž q " i . s q " i m ⭈⭈⭈ m q " i , where K i stands for q iy iq 1 for i s 1, . . . , m y 1. Let ᒄ s gl m 1 [ ⭈⭈⭈ [ gl m r be a Levi subalgebra of gl m and Uq Ž ᒄ . the subalgebra of Uq Ž gl m . associated to ᒄ. By the restriction, Uq Ž ᒄ . acts on VKmn, which is again denoted by . Let s Ž Ž1., . . . , Ž r . . be a multipartition consisting of r-tuples of partitions Ž i.. We denote by < < s Ý ris1 < Ž i. < the size of , where Ž i. s i Ž Ž1i., . . . , Žki. . is a partition of < Ž i. < s Ý kjs1 Žj i.. Let Pn, r be the set of i Ž1. Ž r . multipartitions s Ž , . . . , . such that < < s n. It is known by wAKx that all the irreducible representations of HK are Žup to isomorphism. parametrized by the set Pn, r . We denote by Z the irreducible HK-module corresponding to g Pn, r . On the other hand, irreducible representations of Uq Ž ᒄ . occurring in VKmn are parametrized by the set ⌳ m 1 , . . . , m r of multipartitions g Pn, r such that l Ž Ž i. . F m i . ŽHere for a partition s Ž 1 , . . . , k ., we put l Ž . s k.. We denote by V the irreducible Uq Ž ᒄ .-module corresponding to g ⌳ m 1 , . . . , m r. Under this setting, the following Schur᎐Weyl reciprocity holds between Uq Ž ᒄ . and HK . THEOREM 2.4 wSS, Theorem 4.1x. Ž HK . and ŽUq Ž ᒄ .. are mutually the full centralizers of each other; i.e., we ha¨ e
Ž HK . s End Uq Ž ᒄ . VKmn ,
Ž Uq Ž ᒄ . . s End HK VKmn .
Moreo¨ er, the HK m Uq Ž ᒄ .-module VKmn is decomposed as VKmn ,
[
g⌳ m1 , . . . , m r
Z m V .
ARIKI ᎐ KOIKE ALGEBRAS
823
One can show that Si and j commute with the action of Uq Ž ᒄ . Žcf. wSS, Lemma 3.4x.. Hence as a corollary to Theorem 2.4, we have COROLLARY 2.5. The operators Si Ž2 F i F n., j Ž1 F j F n. are contained in Ž HK .. 2.6. We fix the set of standard generators t 1 , s2 , . . . , sn4 of W, where t 1r s 1 and s2 , . . . , sn are standard generators of the symmetric group ᑭ n under the isomorphism W , ᑭ n h Ž⺪rr⺪. n. We put t i s si ⭈⭈⭈ s2 t 1 s2 ⭈⭈⭈ si for i s 2, . . . , n. Then t 1 , . . . , t n are generators of Ž⺪rr⺪. n, and any element w g W can be written in a unique way as w s t 1c1 ⭈⭈⭈ t nc n , where g Sn , and c i are integers such that 0 F c i F r y 1. Let V s [Vi with Vi s ⺓ m i . We fix a basis E s ¨ 1 , . . . , ¨ m 4 of V in a similar way as in 2.2. We fix hereafter a primitive r th root of unity in ⺓. The group W acts on the n-fold tensor space V mn ; each si acts as i as in 2.2, and t j acts as jy1. nyj. idmŽ m wt m idmŽ , where wt is a linear map on V defined by wtŽ ¨ j . V V bŽ j. s ¨ j . We denote this representation of W by . Let 0 : R ª ⺓ be the specialization homomorphism defined by 0 Ž q . s 1, 0 Ž u i . s i. Let ⺓W be the group algebra of W over ⺓. By the specialization 0 , one obtains ⺓ mR H , ⺓W. Moreover, via the specialization ⺓ mR V mn , V mn, 1 m Ti and 1 m Si coincide with Ž si . on V mn, and 1 m j coincides with Ž t j .. Note that it is known by wAKx that the subalgebra of H generated by a2 , . . . , a n4 coincides with the Hecke algebra Hn associated to ᑭ n . Then one can define an element a for any g ᑭ n as the product a s a i1 ⭈⭈⭈ a i r according to the reduced expression s si1 ⭈⭈⭈ si r. We denote by T , for g S n , the operator Ž a . on V mn. Now assume that m i G n for i s 1, . . . , n. Then it is known wSS, Proposition 5.3x that the representations and are faithful. In particular, in this case, Ž w . ¬ w g W 4 are linearly independent operators on V mn. For each w s t 1c1 ⭈⭈⭈ t nc n g W, we define an operator Tw on V mn by Tw s 1c1 2c 2 ⭈⭈⭈ nc n T . Then Tw is mapped, by the specialization 0 , to Ž t 1c1 ⭈⭈⭈ t nc n . s Ž w .. Hence by using a similar argument as in the proof of Proposition 5.3 in wSSx, we see that Tw ¬ w g W 4 are linearly independent over R. Since dim K Ž HK . s < W <, we have the following. PROPOSITION 2.7. Assume that m i G n for 1 F i F r. Then Tw ¬ w g W 4 gi¨ es rise to a basis of Ž HK .. 3. A NEW PRESENTATION OF HK 3.1. In this section, we assume that m i G n for each i. In the remainder of this paper, we use the following notation. Let M be an R-module. For any commutative algebra R⬘ over R, we write MR⬘ s R⬘ mR M.
824
TOSHIAKI SHOJI
Let V s [Vi as in 2.2. We write the basis Ei of Vi as ¨ jŽ i. ¬ 1 F j F m i 4 , by preserving the orders. Let U be the R-submodule of V mn spanned by 1 . m ⭈⭈⭈ m ¨ Ž c n . , with g ᑭ ¨Ž cŽ1. Ž n. n and 1 F c i F r for i s 1, . . . , n. Then U is a free R-module of rank < Wn, r <. Note that U is H-stable. In fact, it is clear that U is stable by Ti and Si for i s 2, . . . , n. Since U is stable by 1 , U is also stable by T1. Hence it is stable by H . Let us define an element ¨ 0 g U by r
¨0 s
ž
r
Ý ¨ nŽ i . 1
i1s1
m
/ ž
r
Ži . Ý ¨ ny1 2
i 2s1
/
m ⭈⭈⭈ m
ž
Ý
i n s1
/
¨ 1Ž i1 . .
We have the following lemma. LEMMA 3.2. The HK-module UK is generated by ¨ 0 . In particular, UK is isomorphic to the regular HK-module. Proof. We consider the W-module V mn. We write the basis Ei of Vi as ¨ jŽ i. ¬ 1 F j F m i 4 as above, and define a subspace U of V mn in a similar way as above by replacing ¨ jŽ i. by ¨ jŽ i.. Then U is a ⺓W-submodule of V mn. We also define an element ¨ 0 g U in a similar way. We claim that Ž3.2.1. The ⺓W-module U is generated by ¨ 0 . In order to show the claim, it is enough to see that ⺓W¨ 0 contains all Ž i. 1 . m ⭈⭈⭈ m ¨ Ž c n . . Put f s Ý r the vectors of the form ¨Ž cŽ1. Ž n. j is1 ¨ j . We consider a the action of t j on ¨ 0 s f n m ⭈⭈⭈ m f 1. It only affects the jth component of Ž i. ¨ 0 . The jth component of t ja ¨ 0 is equal to Ý ris1 ai ¨ nyjq1 for a s 0, . . . , r y 1. It follows that Ž i. f n m ⭈⭈⭈ m f nyjq2 m ¨ nyjq1 m f nyj m ⭈⭈⭈ m f 1 g ⺓W¨ 0 ,
for i s 1, . . . , r. Repeating this procedure for j s 1, . . . , n, we see that
¨ nŽ c1 . m ⭈⭈⭈ m ¨ 1Ž c n . is contained in ⺓W¨ 0 for any 1 F c i F r. Then by apply1 . m ⭈⭈⭈ m ¨ Ž c n . g ⺓W¨ . ing g ᑭ n on these vectors, we conclude that ¨Ž cŽ1. Ž n. 0
The claim follows. Now Ž3.2.1. implies that the set w¨ 0 ¬ w g Wn, r 4 is linearly independent over ⺓. Let Tw ¬ w g W 4 be the basis of HK given in 2.6. Then by the specialization 0 , Tw¨ 0 is mapped to w¨ 0 for each w g W. Hence we see that the set Tw¨ 0 ¬ w g W 4 is linearly independent over R. In particular, dim K HK ¨ 0 s < W <. Since HK ¨ 0 ; UK , we conclude that HK ¨ 0 s UK . The lemma is proved. 3.3. We write ¨ 0 as ¨ 0 s f n m ⭈⭈⭈ m f 1 , where f j s Ý ris1 ¨ jŽ i.. The action of ja on ¨ 0 affects only the jth factor f nyjq1 , which is mapped to Ž i. Ý ris1 u ia ¨ nyjq1 . Let A be the matrix of degree r whose ab-entry is equal to
ARIKI ᎐ KOIKE ALGEBRAS
825
u ba for 1 F b F r, 0 F a F r y 1; i.e., A is the usual Vandermonde matrix. Thus ⌬ s det A is the Vandermonde determinant, ⌬ s Ł i ) j Ž u i y u j .. We can write the inverse of A as Ay1 s ⌬y1 B, where B s Ž h b aŽ u.. and h b aŽ u. is a polynomial in ⺪w u1 , . . . , u r x. Under this setting, we have Ž b. f n m ⭈⭈⭈ m f nyjq2 m ¨ nyjq1 m f nyj m ⭈⭈⭈ m f 1 s ⌬y1
ry1
Ý h b a Ž u . ja ⭈ ¨ 0 . as0
Ž 3.31. We now introduce a polynomial Fc Ž X . with a variable X, for 1 F c F r, with coefficients in ⺪wŽ u1 , . . . , u r x by Fc Ž X . s
Ý
h ci Ž u . X i .
Ž 3.3.2.
0FiFry1
The following formula is easily obtained from the definition: Fc Ž u c⬘ . s ⌬ ⭈ ␦cc⬘ .
Ž 3.3.3.
Take j G 2. By repeating the procedure for getting Ž3.3.1. to the action of a jy1 , we have Ž c1 . Ž c2 . f n m ⭈⭈⭈ m f nyjq3 m ¨ nyjq2 m ¨ nyjq1 m f nyj m ⭈⭈⭈ m f 1
s ⌬y2 Fc1Ž jy1 . Fc 2Ž j . ⭈ ¨ 0 .
Ž 3.3.4.
Now by applying Tj on both sides of Ž3.3.4., we have the formula Ž c2 . Ž c1 . f n m ⭈⭈⭈ m f nyjq3 m ¨ nyjq1 m ¨ nyjq2 m f nyj m ⭈⭈⭈ m f 1
¡ T F Ž ¢Ž T y Ž q y q y2
s
~⌬
j
c1
jy1
. Fc 2Ž j . ⭈ ¨ 0
y1
j
y2
. Te . ⌬
Fc1Ž jy1 . Fc 2Ž j . ⭈ ¨ 0
if c1 G c 2 , if c1 - c 2 ,
Ž 3.3.5. where Te stands for the identity operator Žcorresponding to the unit element e g W .. We have the following lemma. LEMMA 3.4.
For j s 2, . . . , n, we ha¨ e
Tj j s jy1Tj q ⌬y2 Tj jy1 s j Tj y ⌬y2 Tj k s k Tj
Ý Ž uc
c1-c 2
Ý Ž uc
c1-c 2
2
2
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . ,
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . ,
Ž k / j y 1, j . .
826
TOSHIAKI SHOJI
In the expression of the formulas, the sum is taken o¨ er all 1 F c1 , c 2 F r such that c1 - c2 . Proof. It is clear that if k / j y 1, j, then Tj k s k Tj . We consider the first and second equalities. By Lemma 3.2, the map HK ª UK , h ¬ hŽ ¨ 0 ., is injective. Hence to prove the equality in the lemma, we have only to evaluate both sides at ¨ 0 . First we compare the values jy1Tj Ž ¨ 0 . and Tj jy1Ž ¨ 0 .. They only affect the j y 1 and jth factors of ¨ 0 s f n m ⭈⭈⭈ m f 1. The j y 1 and jth factor of jy1Tj Ž ¨ 0 . is equal to
Ý
c1Gc 2
Ž c2 . Ž c1 . u c 2¨ nyjq1 m ¨ nyjq2
q
Žc . Žc . Žc . Žc . m ¨ nyjq2 q Ž q y qy1 . u c ¨ nyjq2 m ¨ nyjq1 Ý Ž u c ¨ nyjq1 ., 2
1
2
c1-c 2
1
2
1
and the j y 1 and jth factor of Tj jy1Ž ¨ 0 . is equal to
Ý
c1Gc 2
Ž c2 . Ž c1 . u c1¨ nyjq1 m ¨ nyjq2
q
Žc . Žc . Žc . Žc . m ¨ nyjq2 q Ž q y qy1 . u c ¨ nyjq2 m ¨ nyjq1 Ý Ž u c ¨ nyjq1 .. 2
1
1
c1-c 2
1
2
1
It follows that jy1Tj Ž ¨ 0 . y Tj jy1Ž ¨ 0 . coincides with f n m ⭈⭈⭈ m f nyjq3 m
½ÝŽ
Ž c2 . Ž c1 . u c 2 y u c1 . ¨ nyjq1 m ¨ nyjq2 m f nyj m ⭈⭈⭈ m f 1 .
c1 , c 2
5
A similar formula holds also for Tj and j . Now by using Ž3.3.5., we have
k Tj y Tj k s ⌬y2 q
Ý Ž uc
c1-c 2
2
½ÝŽ c1)c 2
u c 2 y u c1 . Tj Fc1Ž jy1 . Fc 2Ž j .
y u c1 . Ž Tj y Ž q y qy1 . Te . Fc1Ž jy1 . Fc 2Ž j . , Ž 3.4.1.
5
for k s j y 1, j, where s 1 Žresp. y1. if k s j y 1 Žresp. k s j .. Here we note that
j y jy1 s ⌬y2
Ý Ž uc
c1)c 2
2
y u c1 . Fc1Ž jy1 . Fc 2Ž j . y Fc 2Ž jy1 . Fc1Ž j . 4 .
Ž 3.4.2.
ARIKI ᎐ KOIKE ALGEBRAS
827
In fact, we replace the action of Tj on VKmn by j . Then the argument used to prove Ž3.4.1. still valid, and we get the formula, which is obtained from Ž3.4.1. by specializing q s 1 Žand remaining u i unchanged.. Then we have
k j y j k s ⌬y2j
Ý Ž uc
c1)c 2
2
y u c1 . Fc1Ž jy1 . Fc 2Ž j . y Fc 2Ž jy1 . Fc1Ž j . 4 .
Ž 3.4.3. But since jy1 j s j j , we have jy1 j y j jy1 s j Ž j y jy1 .. Hence Ž3.4.2. follows from Ž3.4.3.. Now by virtue of Ž3.4.2., the equality Ž3.4.1. can be rewritten
k Tj y Tj k s Tj Ž j y jy1 . y ⌬y2
Ý Ž uc
c1-c 2
2
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . .
The equalities in the lemma follow from this. Next we show LEMMA 3.5.
For j s 2, . . . , n, we ha¨ e
S j s Tj y ⌬y2 Ž q y qy1 .
Ý
c1-c 2
Fc1Ž jy1 . Fc 2Ž j . .
Proof. It is easily checked that for j s 2, . . . , n, we have S j Ž f n m ⭈⭈⭈ m f 1 . s j Ž f n m ⭈⭈⭈ m f 1 . .
Ž 3.5.1.
Then Ž3.3.5. implies that S j s ⌬y2
Ý
c1Gc 2
q ⌬y2
Tj Fc1Ž jy1 . Fc 2Ž j .
Ý Ž Tj y Ž q y qy1 . Te . Fc Ž jy1 . Fc Ž j . . Ž 3.5.2. 1
c1-c 2
2
As in the proof of Lemma 3.4, we replace Tj by j . Then S j is changed to j . By specializing q s 1 in the above formula, we have 1 s ⌬y2
Ý
Fc1Ž jy1 . Fc 2Ž j . .
c1 , c 2
Substituting Ž3.5.3. into Ž3.5.2., we obtain the required formula.
Ž 3.5.3.
828
TOSHIAKI SHOJI
3.6. Let R1 s ⺪w q, qy1 , u1 , . . . , u r , ⌬y1 x be the Ž⺪-. subalgebra of K generated by R and ⌬y1 . We define an algebra H h over R1 as an associative algebra generated by a2 , . . . , a n and 1 , . . . , n subject to the following relations:
Ž ai y q . Ž ai q qy1 . s 0
Ž 2 F i F n. ,
Ž 3.6.1.
Ž i y u1 . ⭈⭈⭈ Ž i y u r . s 0
Ž 1 F i F n. ,
Ž 3.6.2.
Ž 2 F i F n. ,
Ž 3.6.3.
ai a j s a j ai
Ž < i y j < G 2. ,
Ž 3.6.4.
i j s j i
Ž 1 F i , j F n . , Ž 3.6.5.
a i a iq1 a i s a iq1 a i a iq1
a j j s jy1 a j q ⌬y2
Ý Ž uc
c1-c 2
2
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . ,
Ž 3.6.6. a j jy1 s j a j y ⌬y2 aj k s k aj
Ý Ž uc
c1-c 2
2
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . , Ž 3.6.7.
Ž k / j y 1, j . .
Ž 3.6.8.
Then we have the following theorem. THEOREM 3.7. H h is a free R1-module with rank H h s < W <. Moreo¨ er, K mR 1 H h is isomorphic to HK , and HR 1 is isomorphic to a subalgebra of H h. Proof. For a reduced expression s si1 ⭈⭈⭈ si k g ᑭ n , the product a i1 ⭈⭈⭈ a i k is independent of the choice of the reduced expression Žthis follows from Ž3.6.1., Ž3.6.3., and Ž3.6.4.., which we denote by a . It follows from Ž3.6.6. ᎐ Ž3.6.8. that we see H h is generated over R1 by the elements of the form 1c1 ⭈⭈⭈ nc n a , with g ᑭ n and with integers c k such that 0 F c k F r y 1. We want to show that these elements are actually linearly independent over R1. Let us consider the tensor space VRmn . We note that the corre1 spondence a i ¬ Ti , j ¬ j gives rise to a representation of H h on VRmn . 1 In fact, j acts diagonally on the basis elements on VRmn with eigenvalues 1 u1 , . . . , u r 4 . Hence they satisfy the relations obtained by replacing j by j in Ž3.6.2.. By Lemma 3.4, the relations corresponding to Ž3.6.6. ᎐ Ž3.6.8. are also satisfied. Other relations are trivial. Now let be the thus obtained representation of H h on VRmn . Since j g Ž HK . by Corollary 2.5, we have 1 Ž HKh. ; Ž HK .. By Proposition 2.7, the elements 1c1 ⭈⭈⭈ nc n T 4 are linearly independent over K, and so are linearly independent over R1. This implies that 1c1 ⭈⭈⭈ nc n a 4 are also linearly independent over R1 as asserted.
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The above proof also shows that the representation is faithful. Now we have dim K Ž HK . s < Wn, r <, and is faithful. It follows that HKh , HKh s Ž HK . , HK .
ž /
Next we note, by Lemma 3.5, that S j belongs to Ž H h.. It follows that T1 is contained in Ž H h., and we have Ž HR 1 . ; Ž H h.. Since and are faithful, this implies the last statement. The theorem is now proved. Remark 3.8. Theorem 3.7 asserts that there exist embeddings H ¨ H h ¨ HK . A priori, these embeddings depend on the choice of the representation , i.e., on the space V s [Vi . However, the embeddings are independent of the choice of . In fact, we define an element L i g Ž H . by L1 s T1 and by L i s Ti L iy1Ti inductively for i s 2, . . . , n. Then Lc11 ⭈⭈⭈ Lcnn T 4 gives rise to an R-basis of Ž H . by wAKx. Now by Lemmas 3.4 and 3.5, each Lc11 ⭈⭈⭈ Lcnn T can be written as a linear combination of elements in 1c1 ⭈⭈⭈ nc n T 4 , with coefficients independent of the choice of . It follows that each j is also expressed as a linear combination of elements in Lc11 ⭈⭈⭈ Lcnn T 4 , with coefficients independent of . This shows that the embeddings do not depend on the choice of . By making use of Theorem 3.7, we have the following refinement of Lemma 3.2. COROLLARY 3.9. Let UR 1 s R1 mR U . Then we ha¨ e UR 1 s H h¨ 0 . In particular, UR 1 is isomorphic to H h as left H h-modules. Proof. It is clear that H h¨ 0 is contained in UR 1. On the other hand, the arguments in 3.3 imply that ¨ nŽ c1 . m ⭈⭈⭈ m ¨ 1Ž c n . can be written as ¨ nŽ c1 . m ⭈⭈⭈ m ¨ 1Ž c n . s ⌬yn Fc1Ž 1 . Fc 2Ž 2 . ⭈⭈⭈ Fc nŽ n . ⭈ ¨ 0 .
This implies that UR 1 ; H h¨ 0 , and so we get the required equality, UR 1 s H h¨ 0 . Now the map H h ª UR 1, h ¬ h¨ 0 is surjective. Since it is injective by Lemma 3.2, it gives an isomorphism H h , UR 1.
3.10. We consider an example of H h in the case where n s r s 2. Hence H h is generated by a2 and 1 , 2 . In this case, ⌬ s u 2 y u1 , and we have F1Ž X . s u 2 y X, F2 Ž X . s yu1 q X. Then the commutation formula is given explicitly as a2 2 s 1 a2 y ⌬y1 Ž q y qy1 . Ž 1 y u 2 . Ž 2 y u1 . , a2 1 s 2 a2 q ⌬y1 Ž q y qy1 . Ž 1 y u 2 . Ž 2 y u1 . .
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TOSHIAKI SHOJI
Put C s ⌬y2 Ž q y qy1 .. The operator S2 Žor rather the corresponding element in H h. can be computed by Lemma 3.5: S 2 s a 2 q C Ž 1 y u 2 . Ž 2 y u1 . . Now HK is the Hecke algebra of type B2 , and its basis elements are 1, a1 , a2 a1 , a1 a2 a1 4 and those multiplied by a2 to them from the right. They are expressed as a linear combination of elements of the form 1c1 2c 2 a as follows: a1 s 1 q Cu1 Ž 2 y u 2 . Ž 1 y u1 . a2 , a2 a1 s 2 a2 q Cu 2 Ž 2 y u1 . Ž 1 y u 2 . , a1 a2 a1 s 1 2 a2 q Cu1 u 2 Ž 2 y u1 . Ž 1 y u 2 . q Ž 2 y u 2 . Ž 1 y u1 . 4 a22 . Other basis elements are easily obtained from them. It seems interesting to find an explicit description of 1 , . . . , n in terms of a1 , a2 , . . . , a n . 4. IRREDUCIBLE REPRESENTATIONS OF HK 4.1. In wAKx, Ariki and Koike constructed an irreducible representation of HK corresponding to g Pn, r on the space spanned by the standard tableaux of shape . ŽFor the definition of the standard tableau of shape , see the next section.. In this section we shall give an alternate construction of irreducible representations of HK , which is more like the standard construction of irreducible representations of W in terms of induced representations from certain subgroups. 4.2. For s Ž Ž1., . . . , Ž r . . g Pn, r , we set < Ž i. < s n i , and pi s Ýiks1 n k . Let ᑭ Žni. be the subgroup of ᑭ n generated by s j such that piy1 q 1 - j F pi . Then ᑭ Žni. , ᑭ n i , and we define a parabolic subgroup ᑭ nŽ . of ᑭ n by Ž2. Žr. ᑭ n Ž . s ᑭ Ž1. n = ᑭ n = ⭈⭈⭈ = ᑭ n .
Let W Ž . be the subgroup of W obtained as the semidirect product of ᑭ nŽ . with Ž⺪rr⺪. n. Then W Ž . can be written as W Ž . s W Ž1. = W Ž2. = ⭈⭈⭈ = W Ž r . , where W Ž i. is the subgroup of W generated by ᑭ Žni. and t j such that piy1 - j F pi . We have W Ž i. , Wn i , r . Let Hn be the Hecke algebra over R associated to ᑭ n as given in 2.6. We denote by HnŽ i. the subalgebra of Hn generated by a j with piy1 q 1 - j
ARIKI ᎐ KOIKE ALGEBRAS
831
F pi . Thus HnŽ i. is isomorphic to the Hecke algebra Hn i . One can define a Žparabolic. subalgebra HnŽ . of Hn by Hn Ž . s HnŽ1. m HnŽ2. m ⭈⭈⭈ m HnŽ r . . We define an R1-algebra H Ž . as the subalgebra of H h generated by HnŽ . and 1 , . . . , n . Then it follows from 3.6 that H Ž . can be written as H Ž . s H
Ž1.
mH
Ž2.
m ⭈⭈⭈ m H
Žr.
,
where H Ž i. is the subalgebra of H h generated by HnŽ i. and j such that piy1 - j F pi . We note that H Ž i. , Ž Hn i , r . h. Set Hn,Ž i.K s K m HnŽ i.. We denote by ZŽ i. an irreducible Hn,Ž i.K-module corresponding to a partition Ž i. of n i under the identification HnŽ i. , Hn i . Then ZŽ i. can be extended to an irreducible HKŽ i.-module, where j g HKŽ i. acts on ZŽ i. as the scalar multiplication by u i . To see this, it is enough to show that the relations Ž3.6.1. ᎐ Ž3.6.8. hold once the j are replaced by the scalar multiplication u i , for piy1 - j F pi . The relations are trivial except the case Ž3.6.6. or Ž3.6.7.. Since Fc1Ž u i . Fc 2Ž u i . s 0 for c1 / c 2 by Ž3.3.3., the right hand side of Ž3.6.6. is equal to u i a j when j and jy1 are replaced by u i . This shows that Ž3.6.6. also holds. The verification of Ž3.6.7. is similar. We can now define an H Ž .K-module Z 0 as Z 0 s ZŽ1. m ZŽ2. m ⭈⭈⭈ m ZŽ r . .
Ž 4.2.1.
We obtain an induced module HK mH Ž . K Z0 . Then we have the following theorem. THEOREM 4.3. For each g Pn, r , HK mH Ž . K Z 0 is isomorphic to the irreducible HK-module Z . Proof. We shall construct HK mH Ž . K Z 0 as an HK-submodule of VKmn. i By the Schur᎐Weyl reciprocity between Hn i , K and Uq Ž gl m i . on Vi,mn K , Ž i. mn i Hn, K-module ZŽ i. can be realized as the subspace Hn i , K ¨Ž i. of Vi, K , where ¨Ž i. is a highest weight vector for an irreducible Uq Ž gl m i .-module corresponding to Ž i.. Then HnŽ .K-module Z 0 can be realized inside of VKmn, and it is written as Z 0 s HnŽ .K ¨ , where ¨ s ¨Ž1. m ¨Ž2. m ⭈⭈⭈ m ¨Ž r . g Z 0
is a highest weight vector in VKmn for an irreducible Uq Ž ᒄ .-module corresponding to g ⌳ m 1 , . . . , m r. Now the action of j on V mn is given by j . Then it is easy to see that j acts on Z 0 as a scalar multiplication by u i for piy1 - j F pi . Hence Z 0 can be extended to an H Ž .K-module which coincides with the one given in Ž4.2.1.. Let Zˆ be the HK-submodule of VKmn generated by Z 0 . Then Zˆ can be expressed as Zˆ s HK ¨ , and by
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TOSHIAKI SHOJI
the Schur᎐Weyl reciprocity ŽTheorem 2.4., Zˆ is isomorphic to Z . On the other hand, by using the defining relations in 3.6 for H h, we see that HK is a free right H Ž .K-module with basis a w ¬ w g ᑭ nrᑭ nŽ .4 . But dim K Zˆ s dim K Z is equal to the dimension of the irreducible W-module corresponding to , which is given by r
ᑭ nrᑭ n Ž .
Ł dim K ZŽ i. s
is1
ᑭ nrᑭ n Ž . dim K Z 0 .
This implies that Zˆ , HK mH Ž . K Z 0 , and the theorem follows. 5. ORTHOGONAL PRIMITIVE IDEMPOTENTS 5.1. In wGx, Gyoja constructed a partition of unity, i.e., a complete system of orthogonal primitive idempotents in Hn, K which is specialized to a partition of unity in ⺓ ᑭ n . In this section, we generalize his result to the case of HK . A multipartition may be regarded as an r-tuple of Young diagrams. An r-tuple ⺣ s Ž S Ž1., . . . , S Ž r . . of tableaux is called a standard tableau of shape if each j Ž1 F j F n. occurs exactly once as an entry in one tableau in ⺣, and each S Ž i. is a standard tableau of shape Ž i.. For a g Pn, r , we define an r-tuple of standard tableau ⺣qs ⺣q Ž . as follows: we insert the numbers 1, 2, . . . , n in the r-tuple of Young diagrams by the following rule. First, insert the numbers 1, 2, . . . , n1 in the Young diagram of Ž1. in order along the consecutive rows in Ž1., from left to right, and from top to bottom, and then insert the numbers n1 q 1, . . . , n1 q n 2 in the Young diagram of Ž2. in the same way, and so on, until the last partition Ž r .. Thus ⺣q is defined. Next we define ⺣ys ⺣y Ž . in a similar way as ⺣q, but by using columns instead of rows in the Young diagrams Ž1., . . . , Ž r .. Ž1. Ž r .. We write ⺣ "s Ž S " , . . . , S" . Let Ž i. s Ž Ž1i., Ž2i., . . . . be a partition of Ž i. n1. Then Sq determines an embedding of ᑭ Ž1i. = ᑭ Ž2i. = ⭈⭈⭈ into ᑭ Žni., Ž i. which is the row stabilizer of Sq in ᑭ Žni.. We denote by ᑭ Žn,i.q the thus obtained Žparabolic. subgroup of ᑭ Žni.. Similarly, we denote by ᑭ Žn,i.y the Ž i. parabolic subgroup of ᑭ Žni. obtained as the column stabilizer of Sy . We set Ž2. Žr. ᑭ n , "s ᑭ Ž1. n , " = ᑭ n , " = ⭈⭈⭈ = ᑭ n , " .
Ž 5.1.1.
Then ᑭ n, " are parabolic subgroups of ᑭ n . We now define subgroups W"s W" Ž . of W by W"s ᑭ n, "h Ž⺪rr⺪. n. Then W" can be written as W"s W"Ž1. = W"Ž2. = ⭈⭈⭈ = W "Ž r . ,
Ž 5.1.2.
where W"Ž i. , ᑭ Žn,i. " h Ž⺪rr⺪. n i is a subgroup of W generated by ᑭ Žn,i. "
ARIKI ᎐ KOIKE ALGEBRAS
833
and t j with piy1 - j F pi . Next we consider the Hecke algebra version of these subgroups. Let Hn, " be the parabolic subalgebra of Hn which is specialized to ᑭ n, " by 0 . Then Hn, " can be written as Ž2. Žr. Hn , "s HnŽ1. , "m Hn , "m ⭈⭈⭈ m Hn , " ,
Ž 5.1.3.
where Hn,Ž i." is the subalgebra of Hn, " which is specialized to ᑭ Ž"i. . We now define an R1-algebra H " as the subalgebra of H h generated by Hn, " and 1 , . . . , n . Then it follows from 3.6 that H " can be written as H "s H "Ž1. m H "Ž2. m ⭈⭈⭈ m H "Ž r .,
Ž 5.1.4.
where H "Ž i. is the subalgebra of H h generated by Hn,Ž i." and j with piy1 - j F p. 5.2. In the construction of irreducible representations of HK given in Section 4, we consider the special case where s Ž Ž1., . . . , Ž r . . is such that Ž i. is a partition Ž n. of n, and other Ž j. are empty partitions. Then H Ž . s H h, and so Z s Z 0 gives the irreducible representation of HK . It follows that Z is a one dimensional representation of HK , whose character will be denoted by qŽ i.. Then we have
qŽ i. Ž a j . s q
Ž 2 F j F n. ,
qŽ i. Ž j . s u i
Ž 1 F j F n. .
Ž 5.2.1.
Next we consider the case where is such that Ž i. is a partition Ž1n . of n, and other Ž j. are empty partitions. Then again H Ž . s H h, and Z s Z 0 is a one dimensional representation, whose character will be denoted by qŽ i.. Then we have
qŽ i. Ž a j . s yqy1
Ž 2 F j F n. ,
qŽ i. Ž j . s u i
Ž 1 F j F n. .
Ž 5.2.2.
The characters qŽ i. and qŽ i. are realized over R. We denote by 1Ž i. Žresp. 1Ž i. . the linear character of W obtained from Ž i. Ž q resp. qŽ i. . by the specialization 0 : R ª ⺓. Note that 1Ž r . coincides with the identity character, and 1Ž1. coincides with the character det V , which associates to w g W the determinant det V Ž w . on V. Remark 5.3. The characters qŽ i. and qŽ i. are also expressed by using the standard generators a1 , . . . , a n4 . We have qŽ i. Ž a1 . s qŽ i. Ž a1 . s u i . In fact, by Ž3.3.3., the formula in Lemma 3.5 shows that S j coincides with Tj once jy1 and j are replaced by a scalar multiplication u i . This implies,
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TOSHIAKI SHOJI
by Ž2.2.1., that T1 acts as 1 on the representation space of qŽ i. or qŽ i. inside of VKmn. The assertion follows from this. Ž i. 5.4. Let U be as in 3.1. For i s 1, . . . , r, we define elements ¨ q and in U by
Ž i. ¨y
Ž i. ¨q s
Ž i. ¨y s
Ý
q lŽ . T Ž ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. . ,
Ý
Ž yq . yl T Ž ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. . ,
gᑭ n
Ž .
gᑭ n
Ž 5.4.1.
Ž i. Ž Ž i. . where l Ž . denotes the length of g ᑭ n . Then ¨ q resp. ¨ y is a Ž i. Ž Ž i. . non-zero vector affording the linear character q resp. q of H h, Ž i. respectively. In fact, by the specialization 0 , ¨ q is mapped to an element Ž i. Ž i. mn Ž i. Ý g ᑭ n Ž ¨ n m ⭈⭈⭈ m ¨ 1 . in V which is clearly non-zero. Hence ¨ q is Ž i. also non-zero. Similarly ¨ y is non-zero. Now put
aqs
Ý
gᑭ n
q lŽ . a
ays
Ž . Ý Ž yq . yl a .
gᑭ n
Then it can be checked that a j aqs qaq and a j ays yqy1 ay for j s Ž i. Ž i. Ž i. Ž i. 2, . . . , n. Hence we have Tj ¨ q s q¨ q and Tj ¨ q s yqy1 ¨ y . On the other Ž i. mn Ž i. Ž i. hand, since ¨ q is contained in Vi , we have j ¨ q s u i ¨ q . Similarly, we Ž i. Ž i. Ž i. Ž Ž i. . have j ¨ y s u i¨ y . It follows that ¨ q resp. ¨ y generates an H h-module affording the character qŽ i. Žresp. qŽ i. .. By the specialization 0 , we have elements ¨ Ž"i. g U , which are given as Ž i. ¨q s
Ž i. ¨y s
Ý
Ž ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. . ,
Ý
Ž y1. l Ž ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. . .
gᑭ n
gᑭ n
Ž .
Ž 5.4.2.
Let i be the character of Ž⺪rr⺪. n which is the restriction of 1Ž i. or 1Ž i. on W Ži.e., the character such that i Ž t j . s i for j s 1, . . . , n.. Then ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. is the image of ¨ 0 under the projection from V mn to the isotopic subspace for i . It follows that one can write ¨ Ž"i. s e Ž"i. ¨ 0 , where e Ž"i. are idempotents in ⺓W, up to non-zero scalar, affording the character 1Ž i. or 1Ž i., which are given as follows: Ž i. eq s ryn
Ý
1Ž i. Ž wy1 . w,
wgW Ž i. ey
yn
sr
Ý wgW
1Ž i. Ž wy1 . w.
Ž 5.4.3.
ARIKI ᎐ KOIKE ALGEBRAS
835
We now return to the case of H h. Since ¨ Ž"i. g UR 1, it follows from Ž i. Corollary 3.9 that there exist e Ž"i. g H h such that ¨ Ž"i. s e Ž"i. ¨ 0 . Then H heq h Ž i. h Ž i. Žresp. H ey . is an H -module affording the linear character q Žresp. Ž i. Ž i. Ž i. qŽ i. .. In particular, we have Ž e Ž"i. . 2 s ␥ " e " for some ␥ " g R1. We note Ž i. that ␥ " / 0. In fact, by the specialization 0 : R1 ª ⺓, e Ž"i. are mapped to the elements e Ž"i. in ⺓W, which are idempotents up to scalar from the Ž i. .y1 Ž i. Ž i. previous remark. Hence ␥ " is non-zero, and Ž␥ " e " give rise to h idempotents in H . Since e Ž"i. are idempotents up to scalar, e Ž"i.H h turn out to be right H h-modules affording the character qŽ i. or qŽ i.. In particular, for each w g W, we have Ž i. Ž i. Ž i. a w eq s eq a w s qŽ i. Ž a w . eq , Ž i. Ž i. Ž i. a w ey s ey a w s qŽ i. Ž a w . ey .
Ž 5.4.4.
5.5. Now the linear characters qŽ i., qŽ i. on Hn ih, r give rise to linear characters on H Ž i. under the isomorphism Hn ih, r , H Ž i.. We denote again by qŽ i. or qŽ i. their restriction to H "Ž i.. Let us define a linear character q Žresp. q . on Hq Žresp. Hy . by
q s qŽ1. m qŽ2. m ⭈⭈⭈ m qŽ r . , q s qŽ1. m qŽ2. m ⭈⭈⭈ m qŽ r . .
Ž 5.5.1.
Fixing i for the moment, we write Ž i. s s Ž 1 , 2 , . . . .. According to the decomposition WqŽ i. , W1 , r = W 2 , r = ⭈⭈⭈ , HqŽ i. is decomposed as Ž i. Hq , H1h, r m H 2h, r m ⭈⭈⭈ .
Ž 5.5.2.
Ž i. Then the linear character q Ž i. on Hq is decomposed as qŽ i. s q,Ž i.1 m q,Ž i.2 Ž i. Ž i. Ž i. m ⭈⭈⭈ , where q, j is nothing but q on Hhj, r as given in 5.2. Let eq, j be h Ž i. Ž . the idempotent up to scalar in H j, r corresponding to q, j as given in 5.4. Ž i. Ž i. Then under the isomorphism Ž5.5.2., eq, 1 m eq, 2 m ⭈⭈⭈ gives rise to an Ž i. idempotent Žup to scalar. in Hq corresponding to qŽ i., which will be Ž i. denoted by eq . In a similar way, according to the decomposition
HyŽ i. , HU1h, r m HU2h, r m ⭈⭈⭈ , where * s Ž U1 , U2 , . . . . is the dual partition of , qŽ i. is decomposed as Ž i. qŽ i. s q,Ž i.1 m q,Ž i.2 m ⭈⭈⭈ , and we define an idempotent Žup to scalar. ey Ž i. Ž i. Ž i. Ž i. corresponding to q as ey s ey, 1 m ey, 2 m ⭈⭈⭈ . We now define elements e " in H "s H "Ž1. m ⭈⭈⭈ m H "Ž r . by Ž2. Žr. e "s e Ž1. " m e " m ⭈⭈⭈ m e " .
Ž 5.5.3.
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TOSHIAKI SHOJI
Then Ž e " . 2 s ␥ " e " with non-zero elements ␥ "g R1. Hence ␥y1 " e " are idempotents in H " affording the linear character q or q . Moreover, if we define e "g ⺓W " by Ž2. Žr. e "s e Ž1. " m e " m ⭈⭈⭈ m e " ,
where e Ž"i. are idempotents in ⺓W Ž i. affording linear characters 1Ž i. or 1Ž i., then e " are idempotents Žup to scalar. affording the linear characters 1 or 1 of W". Under the specialization 0 , e " are mapped to e "g ⺓W". ŽThe linear characters 1 , 1 are defined in a similar way as in the case H ".. We note that H heq, H h mHq q , and H hey, H h mHy q . In fact this follows, by the specialization 0 , from the corresponding facts for ⺓W. ŽSee the proof of Theorem 4.3.. We have the following lemma. LEMMA 5.6.
Let e " be as in 5.5. Then dim K Hom HK Ž HK e " , HK e . . s 1.
Proof. For partitions s Ž 1 , 2 , . . . . and s Ž 1 , 2 , . . . . of n, we define a partial order G by the condition that Ýiks 1 k G Ýiks1 k for i s 1, 2, . . . . Then for s Ž Ž1., . . . , Ž r . ., s Ž Ž1., . . . , Ž r . . g Pn, r , we define a partial order G by the condition that < Ž i. < s < Ž i. < and that Ž i. G Ž i. for each 1 F i F r. In order to prove the lemma, it is enough to show the following statement. Ž5.6.1. If an irreducible representation Z occurs in Hk eq Žresp. HK ey ., then G Žresp. F .. Moreover, the multiplicity of Z in HK e " is equal to one. We show Ž5.6.1.. First consider the q case. Let ˜qŽ i. be the linear character of Hn,Ž i.q obtained by restricting qŽ i. from HqŽ i. to Hn,Ž i.q. Then HKŽ i. mHq,Ž i.K qŽ i. is isomorphic to the Hn,Ž i.K-module Hn,Ž i.K mH n,Ž i.q, K ˜qŽ i., extended to the HKŽ i.-module by defining the action of j Ž i. g H by the scalar multiplication u i . In fact, this is easily checked by realizing the corresponding representation of Hn i , K inside of V mn i as in the proof of Theorem 4.3. Let be a partition of n i . Now it is well known that an irreducible Hn,Ž i.K-module Z occurs in Hn,Ž i.K mH n,Ž i.q, K ˜qŽ i. only when G Ž i., and ZŽ i. occurs exactly once. The assertion Ž5.6.1. follows from this by virtue of Theorem 4.3. The y case is dealt with similarly, by using the fact that Z occurs in Hn,Ž i.K mH n,Ž i.y, K ˜qŽ i. only when F Ž i. and ZŽ i. occurs exactly once. ŽHere ˜qŽ i. is the character of Hn,Ž i.y obtained as the restriction of qŽ i. to Hn,Ž i.y.. Thus Ž5.6.1. is proved, and the lemma follows.
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5.7. Let us take non-zero intertwining operators f "g Hom H K Ž HK e . , HK e " . . Then by Lemma 5.6, f " is unique up to scalar, and the image of f " is isomorphic to the irreducible HK-module Z . We determine the operator fq explicitly. Since Ž␥ " .y1 e " are idempotents affording the characters q or q of H ", K , one can write y1 y1 fq Ž ey . s ␥q ␥y ey fq Ž ey . eq .
Let X be a complete set of representatives in W for the double cosets Wy_ WrWq. Then we have Ž5.7.1. The set ey a w eqg HK ¬ w g X, ey a w eq/ 04 gives a basis of the space ey HK eq. In fact, by using the defining relations in 3.6, one can check that a w 1 w w 2 g Hy a w Hq for w 1 g Wy, w 2 g Wq. Hence by Ž5.4.4., ey HK eq is generated by ey a w eq¬ w g X 4 . Note that by the specialization 0 , ey H heq is mapped to ey ⺓Weq, whose basis is given by ey weq¬ w g X, ey weq/ 04 . In particular, ey a w eq/ 04 are linearly independent over R1. Ž5.7.1. follows from this. We have Wy_ WrWq, ᑭ n, y_ ᑭ nrᑭ n, q, and ᑭ n, " are parabolic subgroups in ᑭ n . Hence we can choose X as the subset of ᑭ n consisting of distinguished representatives for the double cosets ᑭ n, y_ ᑭ nrᑭ n, q; i.e., we fix X as the set of g ᑭ n such that is a minimal length element in the double coset ᑭ n, y ᑭ n, q. Thus fq Ž ey . g ey HK eq can be written as fq Ž ey . s Ý c ey a eq gX
with c g K. For standard tableaux ⺣1 and ⺣ 2 of shape , we denote by Ž⺣1 , ⺣ 2 . the element in ᑭ n which maps each entry in ⺣ 2 to the corresponding entry in ⺣ 1. Put 0 s Ž⺣y, ⺣q .. We note that 0 g X. In fact, since 0 g ᑭ nŽ ., we are reduced to the case where r s 1, and the claim is easily checked in this case Žcf. wG, 2x.. We have the following lemma. LEMMA 5.8. Assume that g X. Then ey a eqs 0 except the case where s 0 . In particular, fq Ž ey . coincides with ey a 0 eq up to a non-zero scalar in K. Proof. First we note that the following formula holds in HK . The proof is straightforward from Ž3.6.6. and Ž3.6.7..
k a eqs a k ⬘ eqq
Ý
⬘-
c ⬘ a ⬘ eq ,
for 1 F k F n, g ᑭ n , where k⬘ s y1 Ž k . and c ⬘ g K.
Ž 5.8.1.
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TOSHIAKI SHOJI
Next we show that ey a eqs 0 unless g ᑭ nŽ . l X. Assume that f ᑭ nŽ .. Then there exists t j g W Ž c1 . and t k g W Ž c 2 . with c1 / c2 such that t j y1 s t k . In view of Ž5.8.1., ey k a eq can be written as ey k a eqs ey a j eqq
Ý
⬘-
c ⬘ ey a ⬘ eq .
In this equation, we may assume that ⬘ g X. It follows, by Ž5.4.4., that we have
Ž u c 2 y u c1 . ey a eqs
Ý
⬘-
c ⬘ ey a ⬘ eq .
But this implies ey a eqs 0, in view of Ž5.7.1.. We now assume that g ᑭ nŽ . l X. Then the assertion of the is easily reduced to the case where r s 1, i.e., the case of the algebra Hn of type A ny1. But this is exactly the case discussed by i.e., he proved in wG, pp. 845᎐846x the corresponding fact for Hn . our assertion follows from his result. Thus the lemma is proved.
lemma Hecke Gyoja; Hence
5.9. Let ⺣ be a standard tableau of shape . For each ⺣, we shall construct an idempotent in HK affording the irreducible representation Z , following the strategy employed in wGx. We set " Ž⺣. s Ž⺣, ⺣ " ., and denote by a " Ž⺣. the elements a for s " Ž⺣.. Since y Ž⺣.y1q Ž⺣. s 0 , the element ay Ž⺣.y1 aq Ž⺣. coincides with a 0 modulo Ž q y qy1 . HK . Hence by Lemma 5.8, we see that ey ay Ž⺣.y1 aq Ž⺣. eq coincides with ey a 0 eq up to a non-zero scalar. By taking fq appropriately according to ⺣, we may assume that fq Ž ey . s ey ay Ž ⺣ .
y1
aq Ž ⺣ . eq .
Ž 5.9.1.
A similar argument also works for fy, and we have fy Ž eq . s eq aq Ž ⺣ .
y1
ay Ž ⺣ . ey .
Ž 5.9.2.
Now by Schur’s lemma, applied for fq Ž HK ey . , Z , one can write fq fy fq Ž ey . s cfq Ž ey . with some c s cŽ⺣. g K. Then by Ž5.9.1. and Ž5.9.2., we have y1 y1 y1 y1 ey ay aq eq aq ay ey ay aq eqs cey ay aq eq , y1 where a "s a " Ž⺣.. If we set eŽ⺣. s ay ey ay1 y aq eq aq , the above formula implies that 2
e Ž ⺣. s c ⭈ e Ž ⺣. .
Ž 5.9.3.
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We claim that Ž5.9.4. c is a non-zero element in K which is mapped to c g ⺓ y 04 via the specialization 0 . Let e "g ⺓W be as in 5.5. By the specialization 0 , the element eŽ⺣. is y1 mapped to an element eŽ⺣. s y ey y1 y q eq q g ⺓W, where "s 2 Ž⺣, ⺣ " .. We have the equality eŽ⺣. s ceŽ⺣. with c g ⺓. Hence in order to show the claim, it is enough to see that eŽ⺣. is an idempotent up to a non-zero scalar. Here we show a more general fact. The following lemma implies the claim Ž5.9.4.. LEMMA 5.10. The element eŽ⺣. is an idempotent in ⺓W up to a non-zero scalar. Moreo¨ er, if ⺣ and ⺣⬘ are standard tableaux of shape such that ⺣ / ⺣⬘, then we ha¨ e eŽ⺣. eŽ⺣⬘. s 0. Proof. The first assertion is easily reduced to the case where r s 1, i.e., to the case where W"s W"Ž i. and ⺣ s S Ž i. is a standard tableau of shape Ž is a partition of n.. Hence e "s e Ž"i. are idempotents Žup to scalar. y1 affording the linear character 1Ž i. or 1Ž i. of W". Then q Wq q , ᑭ n, n y1 n h Ž⺪rr⺪. and y Wy y , ᑭ n, * h Ž⺪rr⺪. , where ᑭ n, Žresp. ᑭ n, * . is the row stabilizer Žresp. the column stabilizer. of S Ž i. in ᑭ n . Let SŽ i. be y1 the linear character of q Wq q obtained from 1Ž i. by the conjugation y1 by q, and similarly the linear character SŽ i. of y Wy y is defined. Ž i. y1 y1 Then " e " " are the idempotents in " ⺓W" " affording the linear character SŽ i. or SŽ i. Žup to scalar.. Then, as in Ž5.4.3., we have yn q eq y1 q sr
Ý
SŽ i. Ž w . w.
wg qW q y1 q
Let i be the linear character of Ž⺪rr⺪. n obtained from 1Ž i. or 1Ž i. as before. If we put eq Ž S . s
Ý
gᑭ n,
,
e0 s ryn
Ý
i Ž w . w,
wg Ž⺪rr ⺪ . n
Ž . Ž . we have q eq y1 q s e 0 eq S s eq S e 0 . A similar argument shows that Ž . Ž . y ey y1 s e e S s e S e , where ey Ž S . s Ý g ᑭ n, *Žy1. lŽ . . This y 0 y y 0 implies that e Ž S . s e0 ey Ž S . ⭈ e0 eq Ž S . s e0 ey Ž S . eq Ž S . ,
Ž 5.10.1.
since e02 s e0 and e0 commutes with e " Ž S .. Here we note that ey Ž S . eq Ž S . is the Young symmetrizer in ᑭ n associated to the standard tableau S, and so is the idempotent up to scalar. Thus the first assertion of Lemma 5.10 follows.
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TOSHIAKI SHOJI
For the second assertion, it is enough to show that eq q Ž ⺣ .
y1
y Ž ⺣⬘ . eys 0.
Ž 5.10.2.
But a similar statement as in Lemma 5.8 holds by replacing eq with ey, and 0 with Ž⺣q, ⺣y .. Hence, by the specialization 0 , we see that eq eys 0 unless g ᑭ n, q Ž⺣q, ⺣y . ᑭ n, y. Since q Ž⺣.y1y Ž⺣⬘. f ᑭ n, q Ž⺣q, ⺣y . ᑭ n, y, we obtain Ž5.10.2.. 5.11. By Ž5.9.4. and by Lemma 5.10, cŽ⺣.y1 eŽ⺣. is an idempotent in HK , which is specialized by 0 to an idempotent cŽ⺣.y1 eŽ⺣. in ⺓W. We note that eŽ⺣. is an idempotent affording the irreducible representation Z since HK a " e " ay1 " , HK e " as HK -modules. In particular, we have eŽ⺣. eŽ⺣⬘. s 0 if ⺣ and ⺣⬘ have different shape. By the specialization 0 , a similar formula holds also for eŽ⺣.. Hence, by combining with Lemma 5.10, we have Ž5.11.1. The set cŽ⺣.4 gives a complete set of mutually orthogonal primitive idempotents in ⺓W, where ⺣ runs over all the standard tableaux of any shape in Pn, r . Returning to the setting for HK , we consider the idempotents eŽ⺣. and Ž e ⺣⬘. in the case where ⺣ and ⺣⬘ are standard tableaux of the same shape. The following lemma is a generalization of Lemma 2.3.1 in wGx. LEMMA 5.12. Let ⺣, ⺣⬘ be standard tableaux of shape such that ⺣ / ⺣⬘. Put y Ž⺣. s 1 1X Ž resp. y Ž⺣⬘. s 2 2X ., where 1 Ž resp. 2 . is the minimal length element in the coset y Ž⺣. ᑭ nŽ . Ž resp. y Ž⺣⬘. ᑭ nŽ .., and 1X , 2X g ᑭ nŽ .. Assume that l Ž 1 . F l Ž 2 . if 1 / 2 , and assume that l Ž 1X . G l Ž 2X . if 1 s 2 . Then we ha¨ e eŽ⺣. eŽ⺣⬘. s 0. Proof. As discussed in 5.10, it is enough to show that eq aq Ž ⺣ .
y1
ay Ž ⺣⬘ . eys 0.
Ž 5.12.1.
Let ⺣ s Ž S Ž1., . . . , S Ž r . .. Then it is easy to see that 1 is written as
1 s
ž
⭈⭈⭈
piy1 q 1
piy1 q 2
⭈⭈⭈
pi
⭈⭈⭈
⭈⭈⭈
k 1Ž i.
k 2Ž i.
⭈⭈⭈
k nŽ i.i
⭈⭈⭈
/
, Ž 5.12.2.
where k 1Ž i. - k 2Ž i. - ⭈⭈⭈ - k nŽ i.i are the letters occurring in the entries of the tableau S Ž i.. This implies that 1 is also the minimal length element in the coset q Ž⺣. ᑭ nŽ ., and so one can write q Ž⺣. s 1 1Y with 1Y g ᑭ nŽ .. Hence we have aq Ž⺣. s a 1 a 1Y , and ay Ž⺣⬘. s a 2 a 2X . We note that, by virtue of Ž5.12.2., there exist standard tableaux ⺣ 0 , ⺣X0 such that 1Y s q Ž⺣ 0 ., 1X s y Ž⺣ 0 . and that 2X s y Ž⺣X0 ..
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First we assume that l Ž 1 . F l Ž 2 . with 1 / 2 . Let Y s 2 g ᑭ n ¬ 4 . Then it is easy to see that ay1 a can be written as F y1 1 1 2 ay1 a 2 s 1
Ý cy ay
Ž cy g K . .
Ž 5.12.3.
ygY
We note that y f ᑭ nŽ .. In fact, if y s 2 g ᑭ nŽ ., then 2 g y1 ᑭ nŽ .. It follows that l Ž y1 . G l Ž 2 ., and by our assumption, l Ž y1 . y1 Ž . G l Ž 1 .. Combined with F y1 1 , we have s 1 . But then 1 ᑭ n s 2 ᑭ nŽ . and this contradicts the assumption that 1 / 2 . Now it follows from Ž5.12.3. that we can write eq aq Ž ⺣ .
y1
ay Ž ⺣⬘ . eys
Ž c g K . ,
Ý c eq a ey
where g ᑭ n y ᑭ nŽ .. A similar argument as in the proof of Lemma y1 5.8 shows that eq a eys 0 unless g ᑭ n, q y1 0 ᑭ n, y, where 0 s y1 Ž⺣q, ⺣y .. Since 0 g ᑭ nŽ ., we get Ž5.12.1.. Next assume that l Ž 1X . G l Ž 2X . with 1 s 2 . Then we have aq Ž ⺣ .
y1
Y a X s a ay Ž ⺣⬘ . s ay1 2 q Ž ⺣0 . 1
y1
ay Ž ⺣X0 . ,
with ⺣ 0 / ⺣X0 . This is essentially the same situation as discussed in wG, Lemma 2.3.1x, and the argument there works well for our case. For the sake of completeness, we give the proof below. For any standard tableau ⺣ such that q Ž⺣. g ᑭ nŽ ., we have l Ž q Ž ⺣ . . q l Ž y Ž ⺣ . . s l Ž y1 0 .. By applying this formula with ⺣ s ⺣ 0 , and by our assumption, we have X l Ž y1 0 . G l Ž q Ž ⺣ 0 . . q l Ž y Ž ⺣ 0 . . .
Ž 5.12.4.
Now aq Ž⺣ 0 .y1 ay Ž⺣X0 . can be written as the form Ý y g Y ⬘ c y a y with c y g K, where Y ⬘ s x y Ž⺣X0 . ¬ x F q Ž⺣ 0 .y1 4 . Since eq a y eys 0 unless y g Ž .y1y Ž⺣X0 . by Ž5.12.4.. ᑭ n, q y1 0 ᑭ n, y, we have eq a y eys 0 if y / q ⺣ 0 X . s l Ž y .. Then y f Assume that y s q Ž⺣ 0 .y1y Ž⺣ 0 . and that l Ž y1 0 y1 y1 ᑭ n, q 0 ᑭ n, y since 0 is the unique minimal length element in this Ž . double coset, and y / y1 0 . Hence 5.12.1 holds also in this case. The lemma is now proved. 5.13. Let T be the set of standard tableaux of shape . We give a total order ⺣ G ⺣⬘ in T , compatible with the property given in Lemma 5.12, i.e., the order satisfying the following properties: assume that ⺣ ) ⺣⬘, and put y Ž⺣. s 1 1X and y Ž⺣⬘. s 2 2X as in Lemma 5.12. Then we
842
TOSHIAKI SHOJI
require that l Ž 1 . F l Ž 2 . if 1 / 2 , and l Ž 1X . G l Ž 2X . if 1 s 2 . Clearly such an order exists. We give a numbering on T as T s ⺣ 1 , ⺣ 2 , . . . , ⺣ f 4 along this order. Then by Lemma 5.12, we see that eŽ⺣ i . eŽ⺣ j . s 0 if i ) j. We define an element ˜ eŽ⺣ i . g HK by y1 y1 y1 ˜e Ž ⺣ i . s Ž 1 y cy1 1 e Ž ⺣ 1 . .Ž 1 y c 2 e Ž ⺣ 2 . . ⭈⭈⭈ Ž 1 y c iy1 e Ž ⺣ iy1 . . c i e Ž ⺣ i . ,
with c i s cŽ⺣ i .. It is proved in Lemma 4.4 in wGx that if eŽ⺣1 ., . . . , eŽ⺣ f .4 is a set of idempotents satisfying the above properties, then ˜ eŽ⺣. < ⺣ g T 4 gives a family of mutually orthogonal idempotents. ŽIt is also shown in wGx that ˜ eŽ⺣. does not depend on the choice of the total order in T .. It is clear that ˜ eŽ⺣. is a primitive idempotent affording the irreducible module Z . It follows from Lemma 5.10 that, by the specialization 0 , ˜ eŽ⺣. is mapped to the idempotent cŽ⺣.y1 eŽ⺣. g ⺓W. Summing up the above arguments, we have the following theorem. THEOREM 5.14. The set ˜ eŽ⺣. g HK ¬ ⺣ g T , g Pn, r 4 gi¨ es a complete family of mutually orthogonal primiti¨ e idempotents in HK . By the specialization 0 , ˜ eŽ⺣. is mapped to the primiti¨ e idempotent ˜ cŽ⺣.y1 eŽ⺣. g ⺓W, which forms a complete family of mutually orthogonal primiti¨ e idempotents in ⺓W.
6. A FROBENIUS FORMULA FOR THE CHARACTERS OF HK 6.1. In this section, we introduce indeterminates x Žj i. Ž1 F i F r, 1 F j F m i . associated to the basis ¨ jŽ i. of V. We also write these indeterminates as x 1 , . . . , x m , corresponding to the basis ¨ 1 , . . . , ¨ m . Let s Ž Ž1., . . . , Ž r . . be a multipartition in Pn, r . Following wM. I. Appendix Bx, we extend the notion of power sum symmetric functions and Schur functions to the case of multipartitions. Recall that is a primitive r th root of unity. We define a ‘‘power sum function’’ P associated to as follows. For each integer k G 1 and i such that 1 F i F r, put r
PkŽ i. Ž x . s
Ý yi j pk Ž x Ž j. . , js1
where pk Ž x Ž j. . denotes the kth power sum symmetric function with respect Ž j. to the variables x 1Ž j., . . . , x m . For a partition Ž i. s Ž Ž1i., Ž2i., . . . , Žki.i . of n i , j Ž . Ž i. we define a function P x by ki
PŽ i. Ž x . s
Ł PŽ i. Ž x . , js1
Ž i. j
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843
and define a function PŽ x . by r
P Ž x . s
Ł P Ž x . . Ž i.
Ž 6.1.1.
is1
We denote by PŽ x . the complex conjugate of PŽ x .. Next we define a ‘‘Schur function’’ S Ž x . associated to by r
S Ž x . s
Ł s Ž x Ž i. . , Ž i.
Ž 6.1.2.
is1
where sŽ i. Ž x Ž i. . denotes the Schur function associated to the partition Ž i. Ž i. with respect to the variables x 1Ž i., . . . , x m . i The set of conjugacy classes in W is parametrized by Pn, r . The correspondence is given as follows. Let w s t 1c1 ⭈⭈⭈ t nc n g W. Now g ᑭ n is written as a product of disjoint cycles. Let z s Ž i1 , i 2 , . . . , i k . be a cycle occurring in the decomposition of . We say that z is of type a if c i1 q c i 2 q ⭈⭈⭈ qc i k ' a Žmod r ., Ž1 F a F r .. Then w belongs to the conjugacy class corresponding to s Ž Ž1., . . . , Ž r . ., if Ž i. is defined as the product of cycles of type i. For each g Pn, r , we fix a representative w of the conjugacy class in W corresponding to as follows. First we put w Ž n, i . s t ni sn sny1 ⭈⭈⭈ s2 . Then w Ž n, i . corresponds to an n-cycle of type i. For a partition s Ž 1 , 2 , . . . . of n, we define w Ž , i . g W by w Ž , i . s w Ž 1 , i . = w Ž 2 , i . = ⭈⭈⭈ under the natural embedding W1 , r = W 2 , r = ⭈⭈⭈ into W. Let s Ž Ž1., . . . , Ž r . . g Pn, r . Under the isomorphism W Ž i. , Wn , r , we may regard i w Ž Ž i., i . as an element in W Ž i.. Then we define w g W Ž . ; W as w s w Ž Ž1. , 1 . w Ž Ž2. , 2 . ⭈⭈⭈ w Ž Ž r . , r . .
Ž 6.1.3.
The above construction of w is generalized as follows. Let ⌺ be the subset of W consisting of w s t 1c1 ⭈⭈⭈ t nc n such that any k-cycle 1 occurring in the decomposition of is of the form 1 s s pqky2 ⭈⭈⭈ s pq1 s p a for some p G 2, and w is a product of the factors t pq ky2 s pqky2 ⭈⭈⭈ s pq1 s p . We write as ⌺ the set of elements of ⌺ which is conjugate to w . Then w g ⌺ is obtained in a similar way as w by using a suitable embedding WŽ1. = WŽ1. = ⭈⭈⭈ into W. 1 ,r 2 ,r Let 1 be the irreducible character of W corresponding to g Pn, r . Then it is known by wM, Appendix B, Ž9.5.x that the formula S s
Ý
g Pn, r
zy1 1 Ž w . P
holds, where z is the order of the centralizer of w in W.
Ž 6.1.4.
844
TOSHIAKI SHOJI
6.2. Let K ⬘ s K Ž x 1 , . . . , x m . be the field of rational functions on K. In the remainder of this paper, we assume that Uq Ž ᒄ ., H , V, etc., are defined over K ⬘. ŽBy abbreviation, we use the notations such as H , V, etc., instead of HK ⬘ , VK ⬘ , etc.. Let Cn, m be the set of m-tuples of integers c i G 0 such that Ýc i s n. Take c s Ž c1 , . . . , c m . g Cn, m . For a Uq Ž ᒄ .-module M, we denote by Mc the weight subspace of M with weight c; i.e., q j acts on Mc by the scalar multiplication by q c j. We consider the H m Uq Ž ᒄ .-module V mn. For each sequence c as above, let Ec be the projection from V mn to the weight subspace Vcmn of V mn. We define an operator D on V mn by Ds
Ý x c Ec , c
where c runs over all the elements in Cn, m , and x c denotes x 1c1 x 2c 2 ⭈⭈⭈ x mc m . Note that weight spaces in V mn are left invariant by the action of H . Hence Ec commutes with the action of H , and so D also commutes with H . We also note that Vcmn coincides with the subspace of V mn generated by ¨ i1 m ¨ i 2 m ⭈⭈⭈ m ¨ i n such that 噛 i k < bŽ i k . s j4 s c j . Let ⺓⬘ s ⺓Ž x 1 , . . . , x m .. In a similar way as above, we regard V as defined over ⺓⬘, and consider the ⺓⬘W m UŽ ᒄ .-module V mn. We define an operator ⌺ c x c Ec on V mn, which we denote also by D, where Ec is the projection from V mn to the weight subspace of weight c. The trace on V mn of elements in W multiplied by D is given as follows. The proof is a straightforward computation. LEMMA 6.3. Ži.
Let w Ž n, i . be as in 6.1. Then we ha¨ e
Tr Ž Dw Ž n, i . , V mn . s
m
Ý bŽ j.i x jn s PnŽ i. Ž x . . js1
Žii.
More generally, for w g ⌺ , we ha¨ e Tr Ž Dw, V mn . s P Ž x . .
6.4. Let p be a primitive idempotent in ⺓W affording the irreducible representation Z . Then we have Tr Ž Dp , V mn . s S Ž x . .
Ž 6.4.1.
In fact, the proof of Lemma 3.7 in Ram wRx can be applied to our case also. By using the argument there, we have Tr Ž Dp , V mn . s < W
Ý wgW
1 Ž w . Tr Ž Dw, V mn . .
ARIKI ᎐ KOIKE ALGEBRAS
845
Since D commutes with the action of W, TrŽ Dw . only depends on the conjugacy class containing w. Hence, by using Lemma 6.3, Žii., we have Tr Ž Dp , V mn . s < W
Ý
g Pn, r
< C < 1 Ž w . P Ž x . ,
where < C < is the number of elements in the conjugacy class C of W corresponding to . The last formula coincides with S Ž x . by Ž6.1.4.. Thus Ž6.4.1. holds. 6.5. We now pass to the setting for H and consider the H m Uq Ž ᒄ .module V mn. Let q be the irreducible character of H corresponding to . We have the following proposition. PROPOSITION 6.6.
For any h g H , we ha¨ e
Tr Ž Dh, V mn . s
Ý
g Pn, r
q Ž h . S Ž x . .
Proof. The proof is done in a similar way as in the proof of Theorem 3.8 in wRx, once we admit Theorem 5.14. By Theorem 5.14, there exists a complete family of mutually orthogonal primitive idempotents pi ¬ g Pn, r , 1 F i F d 4 of H Žhere pi affords the irreducible module Z with d s dim Z . such that each pi is mapped, by the specialization 0 , to a primitive idempotent pi in ⺓W affording the irreducible module Z . By a standard argument Žcf. wR, Ž3.8.x., we can write Tr Ž Dh, V mn . s
Ý h ii Tr Ž Dpi , V mn . ,
Ž 6.6.1.
, i
where h ii is remarked in and so is left
the diagonal element of the matrix of h on Z . Now as wR, Ž3.5.x, TrŽ Dpi . is independent of indeterminates q, u i , invariant by the specialization 0 . Therefore we have Tr Ž Dpi , V mn . s Tr Ž Dpi , V mn . s S Ž x . .
Ž 6.6.2.
The second identity follows from Ž6.4.1.. By substituting Ž6.6.2. into Ž6.6.1., we get the required formula. ŽThere exists a simple, direct proof of Proposition 6.6 without using Theorem 5.14, suggested by the referee, as follows. Since D g ŽUq Ž ᒄ .., by virtue of Theorem 2.4, it is enough to show that TrŽ D, V . s S Ž x .. But Uq Ž ᒄ . , Uq Ž gl m 1 . m ⭈⭈⭈ m Uq Ž gl m r . and according to this isomorphism, V is decomposed as V , VŽ1. m ⭈⭈⭈ m VŽ r . , where VŽ i. is an irreducible Uq Ž gl m i .-module corresponding to the partition Ž i.. Then TrŽ D, V . can be written as r
Tr Ž D, V . s
Ł Tr Ž D Ž i. , V . , Ž i.
is1
846
TOSHIAKI SHOJI
where D Ž i. is the operator on Vimn i defined analogous to D. Using the corresponding result for the case Uq Ž gl m i . wR, Theorem 3.8x, we see that TrŽ D Ž i., VŽ i. . s sŽ i. Ž x Ž i. .. This implies that TrŽ D, V . s S Ž x . by Ž6.1.2., and the proposition follows.. 6.7. For g Pn, r , we define an element a g H imitating the definition of w in 6.1 as follows. First we put aŽ n, i . s ni a n a ny1 ⭈⭈⭈ a2 . Then, for each partition s Ž 1 , 2 , . . . . of n, we define a Ž , i . s a Ž 1 , i . m a Ž 2 , i . m ⭈⭈⭈ under the embedding H1 , r m H 2 , r m ⭈⭈⭈ into H . Let s Ž Ž1., . . . , Ž r . . g Pn, r . We define a g H Ž . ; H by a s a Ž Ž1. , 1 . a Ž Ž2. , 2 . ⭈⭈⭈ a Ž Ž r . , r . , where as in the group case, aŽ Ž i., i . is regarded as an element in H Ž i. via the isomorphism H Ž i. , Hn i , r . More generally, the element a w for w g ⌺ is also constructed in a similar way as above by using a suitable embedding of HŽ1. m HŽ1. m ⭈⭈⭈ into H . 1 ,r 2 ,r In general, for each w s t 1c1 t 2c 2 ⭈⭈⭈ t nc n g W, we set a w s 1c1 2c 2 ⭈⭈⭈ nc n a . Then, a w ¬ w g W 4 gives rise to a basis of H . We note that the above construction of a w for w g ⌺ is compatible with this definition. By the specialization 0 , a w is mapped to w. In particular, a is mapped to w. We shall compute the trace of Da w for w g ⌺ on V mn. The special case where corresponds to an n-cycle of type i is given explicitly. In the following, we give a slightly general formula. THEOREM 6.8.
Let a w s 1c1 2c 2 ⭈⭈⭈ nc n a n a ny1 ⭈⭈⭈ a2 . Then we ha¨ e
Tr Ž Da w , V mn . s
Ý u bŽc i . u bŽc i . 1
2
1
2
cn eŽ I . ⭈⭈⭈ u bŽ Ž q y qy1 . in. q
lŽ I .
x i1 x i 2 ⭈⭈⭈ x i n ,
I
Ž 6.8.1. where the sum is taken o¨ er all the sequences I s Ž i1 , i 2 , . . . , i n . such that 1 F i1 F i 2 F ⭈⭈⭈ F i n , eŽ I . Ž resp. l Ž I .. denotes the number of j such that i j s i jq1 Ž resp. i j - i jq1 .. Proof. The computation of the action of Da n a ny1 ⭈⭈⭈ a2 is completely the same as the proof of Theorem 4.1 in wRx. We have only to note that in this computation, the contribution of the term x i1 x i 2 ⭈⭈⭈ x i n comes from the basis vector ¨ i1 m ¨ i 2 m ⭈⭈⭈ m ¨ i n of V mn, on which k acts as a scalar multiplication by u bŽ i k . . Note that our generator a i g Hn is different from the one in wRx by scalar, and so the expression involving q in our formula is different from wRx.
ARIKI ᎐ KOIKE ALGEBRAS
847
6.9. More generally, we shall compute the trace of Da w for w g ⌺ . The cycle of type i corresponding to Žj i. is expressed as t ki sk sky1 ⭈⭈⭈ sl for some integers l F k such that Žj i. s k y l q 2. Put aŽ Žj i. . s ki a k a ky1 ⭈⭈⭈ a l . Then a can be written as the product of aŽ Žj i. . for various i, j. Note that the operator aŽ Žj i. . affects only the l y 1, l, . . . , k factors of V mn, which are disjoint for distinct Žj i.. Moreover, D is also decomposed according to such a decomposition of factors in V mn. Hence TrŽ Da . can be computed as the product of the traces TrŽ DaŽ Žj i. ... But the latter one coincides with the trace of DaŽ Žj i., i . on the Žj i.-fold tensor space of V. Therefore we have the following formula. PROPOSITION 6.10.
Let w g ⌺ . Then we ha¨ e
Tr Ž Da w , V mn . s
r
l Ž Ži. .
Ł Ł
is1 js1
Tr Da Ž Žj i. , i . , V m j . Ž i.
ž
/
In particular, the trace TrŽ Da w . coincides with the trace TrŽ Da . for any w g ⌺. 6.11. We now introduce a ‘‘monomial function’’ m Ž x . associated to g Pn, r by r
mŽ x . s
Ł m Ž x Ž i. . ,
Ž 6.11.1.
Ž i.
is1
where mŽ i. Ž x Ž i. . is a monomial symmetric function corresponding to the Ž i. partition Ž i. with respect to the variables x 1Ž i., . . . , x m . i Ž i. Ž mn Let qn x; q, u. s TrŽ DaŽ n, i ., V .. Then, thanks to Theorem 6.8, qnŽ i. Ž x; q, u. can be written as qnŽ i. Ž x ; q, u . s
Ý
g Pn, r
ui q nylŽ. Ž q y qy1 .
l Ž .y1
m Ž x . , Ž 6.11.2.
where l Ž . s Ý ris1 l Ž Ž i. . for s Ž Ž1., . . . , Ž r . ., and u denotes u k for the largest number k such that Ž k . is not an empty partition. Let qr Ž x 1 , . . . , x m ; t . be the Hall᎐Littlewood function Žcf. wM, III, 2x., which is defined by q0 Ž x ; t . s 1, m
qr Ž x ; t . s Ž 1 y t .
Ý x ir Ł is1
j/i
x i y tx j xi y x j
Ž r G 1. .
The following formula is obtained from the formula Žb. in wR, Theorem 4.13x:
Ý
&n
q nylŽ . Ž q y qy1 .
l Ž .y1
m Ž x . s
qn q y qy1
qn Ž x ; qy2 . . Ž 6.11.3.
848
TOSHIAKI SHOJI
Let c s Ž n1 , n 2 , . . . , n r . g Cn, r . If s Ž Ž1., . . . , Ž r . . g Pn, r is such that < Ž i. < s n i , we write g c. The following formula expresses the function qnŽ i. Ž x; q, u. in terms of Hall᎐Littlewood functions. PROPOSITION 6.12. qnŽ i. Ž x ; q, u . s
qn
r
q y qy1
u ci Ł qn jŽ x Ž j. ; qy2 . ,
Ý
js1
cg Cn, r
where c g Cn, r is written as c s Ž n1 , n 2 , . . . , n r ., and u c denotes u k for the largest integer k such that n k / 0. Proof. Since u is independent for g c for a fixed c g Cn, r , which is equal to u c , we can write qnŽ i. Ž x ; q, u . s
Ý
u ci
cg Cn, r
Ý
q nylŽ. Ž q y qy1 .
gc
s Ž q y qy1 .
r
ry1
u ci Ł
Ý
= Ž q y qy1 .
l Ž Ž j. .y1
qn q y qy1
Ý
m Ž x .
q n iylŽ
Ž j.
.
js1 Ž j.&n i
cg Cn, r
s
l Ž .y1
mŽ j. Ž x Ž j. .
r
u ic Ł qn jŽ x Ž j. ; qy2 . .
Ý
js1
cg Cn, r
The last equality follows from Ž6.11.3.. Thus the proposition is proved. 6.13. Let qnŽ i. Ž x; q, u. be as in Ž6.11.2.. For a multipartition s Ž Ž1., . . . , Ž r . . g Pn, r , we define a function q˜Ž x 1 , . . . , x m ; q, u. by l Ž Ži. .
r
q˜ Ž x 1 , . . . , x m ; q, u . s
Ł Ł
is1
js1
qŽ i.Žj i. Ž x ; q, u . .
Then by combining Proposition 6.6, Theorem 6.8, and Proposition 6.10, we obtain the following Frobenius type formula for the characters of H , which is a generalization of Theorem 4.14 in wRx. THEOREM 6.14.
For each g Pn, r , we ha¨ e q˜ Ž x ; q, u . s
Ý
g Pn, r
q Ž a . S Ž x . .
Ž 6.14.1.
ARIKI ᎐ KOIKE ALGEBRAS
849
As a corollary, we have the following. COROLLARY 6.15. Let , g Pn, r . Then q Ž a . g R, and by the specialization 0 , q Ž a . is mapped to 1 Ž w .. Proof. It follows from Ž6.11.2. that q˜Ž x; q, u. is an R-linear combination of various m Ž x ., and so is an R-linear combination of S Ž x .. Since Schur functions S Ž x . are linearly independent for g Pn, r , Theorem 6.14 implies that q Ž a . g R. Moreover, the argument used to prove Theorem 6.14 works well for W, and we have q˜ Ž x ; 1, . s
Ý
g Pn, r
1 Ž w . S Ž x . ,
Ž 6.15.1.
where stands for the substitution of , 2 , . . . , r s 1 into u1 , u 2 , . . . , u r . ŽHere q˜Ž x; 1, . coincides with PŽ x ... Hence the second assertion follows by specializing the equation Ž6.14.1. by 0 , and by comparing it with Ž6.15.1..
7. CHARACTER VALUES 7.1. Let l: W ª ⺪ G 0 be the length function of W with respect to the generators t 1 , s2 , . . . , sn4 ; i.e., l Ž w . is the smallest number of generators which are needed to express w as the product of such generators. The function l Ž w . is determined in Bremke and Malle wBMx. Some properties of l Ž w . were studied further in wRSx. In particular, the following is known Žcf. wRS, 1.31x.. For each k Ž1 F k F n. and a Ž0 F a F r y 1., put
n Ž k, a . s
½
skq 1 ⭈⭈⭈ sny1 sn sk ⭈⭈⭈ s2 t a s2 ⭈⭈⭈ sn
if a s 0, if a / 0.
Ž 7.1.1.
Then the set Rn s nŽ k, a. ¬ 1 F i F n, 0 F a F r y 14 gives rise to a complete set of representatives of the left cosets Wn, rrWny1, r . Moreover, we have l Ž ww⬘. s l Ž w . q l Ž w⬘. for w g Rn , w⬘ g Wny1, r . The expression of nŽ k, a. in Ž7.1.1. is a reduced expression with respect to l. The decomposition for Wny 1, r _ Wn, r is also described in wRS, Corollary 1.29x. Combining with the previous result, we have the following double coset decomposition of Wn, r by Wny1, r . Ž7.1.2. Let R˜n s sn , t na ¬ 0 F a F r y 14 . Then any element w g Wn, r can be written as w s w⬘ dw⬙ with w⬘, w⬙ g Wny 1, r , d g R˜n , such that l Ž w . s l Ž w⬘. q l Ž d . q l Ž w⬙ .. Let C s C be the conjugacy class in W corresponding to g Pn, r . We denote by Cmin the set of w g C such that l Ž w . is minimal in C. The following lemma is related to Lemma 2.5 in Geck and Pfeiffer wGPx.
850 LEMMA 7.2.
TOSHIAKI SHOJI
Let C s C for g Pn, r . Then we ha¨ e Cmin l ⌺ / ⭋.
Proof. The proof is done by using a similar argument as in wGP, Proposition 2.3x. Let us take w g C. One can write w s xd n y by Ž7.1.2., with x, y g Wny 1, r and d n g R˜n . Let w⬘ s xy1 wx s d n yx. Then l Ž w⬘. F l Ž w . by the property of the length function in Ž7.12.. Again by Ž7.12., applied to Wny 1, r , we have yx s x⬘d ny1 y⬘ with x⬘, y⬘ g Wny2, r and d ny1 g R˜ny 1. Since elements in R˜n commute with Wny2, r , we see that d n yx s d n x⬘d ny1 y⬘ s x⬘d n d ny1 y⬘. Hence, by a similar argument as before, w⬙ s d n d ny1 y⬘ x⬘ is in C and l Ž w⬙ . F l Ž w .. Repeating this procedure, we can find w 1 s d n d ny1 ⭈⭈⭈ d1 d 0 g C such that l Ž w 1 . F l Ž w ., where d i g R˜i for i G 1 and d 0 g t 1a ¬ 0 F a F r y 14 . The lemma follows from this since w 1 is contained in ⌺ . Remark 7.3. The proof of Theorem 6.14 implies that q Ž a w . takes a constant value for w g ⌺ . Hence q Ž a w . is constant for w g Cmin l ⌺ . In comparing the case of Iwahori᎐Hecke algebras Žsee 7.4 below., it seems interesting to know whether q Ž a w . is constant for w g Cmin . An alternative length function, nŽ w . in the notation in wRSx, associated to the root system of W was introduced in wBMx. One can define the set X Cmin of minimal elements with respect to nŽ w .. It would be also interesting X to consider a similar problem as above for Cmin . 7.4. In wGPx Geck and Pfeiffer have shown, extending the result of wRx for the Hecke algebra of type A n , that in the case of Iwahori᎐Hecke algebras, any character has a constant value on the standard basis Tw for w g Cmin , where Cmin is the set of minimal length element in a given conjugacy class C of the Weyl group, and that the character values at other Tw ⬘ are determined from the values at Tw with w g Cmin . The following result, which is a generalization of Theorem 5.1 in wRx, together with Lemma 7.2, is, in a sense, a counterpart of their result for the case of complex reflection groups GŽ r, 1, n., although our basis of HK is different from the standard basis. This answers, in some sense, a question posed in the last paragraph in Pfeiffer wPx. PROPOSITION 7.5.
For each w g W, there exists Aw s
Ý
g Pn, r
c w , a
with c w, g R1 , such that q Ž a w . s q Ž A w . for any character q of H . In particular, the characters q are determined by a with g Pn, r .
ARIKI ᎐ KOIKE ALGEBRAS
851
Proof. Let ˆ ⌺ be the subset of W consisting of w s t 1c1 ⭈⭈⭈ t nc n such that any k-cycle 1 occurring in the decomposition of is of the form 1 s s pqky2 ⭈⭈⭈ s pq1 s p for some p G 2. Here we introduce a notation. For any w s t 1c1 ⭈⭈⭈ t nc n , we define l 1Ž w . by l 1Ž w . s l Ž .. First we show the following. Ž7.5.1. For each w g W, there exists AXw s
Ý
cw , z a z g H h
ˆ zg⌺
with c w, z g R1 , such that q Ž a w . s q Ž AXw . for any character q of H . Moreover, l 1Ž z . F l 1Ž w . if c w, z / 0. Now take w s t 1c1 ⭈⭈⭈ t nc n g W, and let i be the largest integer such that Ž i . - i y 1. We shall show the existence of AXw by induction on i and reverse induction on Ž i .. Note that if such an i does not exist, then w is contained in ˆ ⌺. Moreover, if i is such a number with Ž i . s n, then we may replace 1, 2, . . . , n4 by 1, 2, . . . , i y 14 . So, let i be such a number with Ž i . / n, and put j s Ž i . q 1. Then j - i and so y1 Ž j . - i. It follows that l Ž s j . - l Ž .. Put ⬘ s s j s j and ⬙ s s j . First assume that l Ž ⬘. ) l Ž ⬙ .. Then a s a j a ⬙ . We have
q Ž a w . s q Ž 1c1 ⭈⭈⭈ nc n a j a ⬙ . s q Ž a j f 1 Ž . a ⬙ . q q Ž f 2 Ž . a ⬙ . , by Ž3.6.6., where f 1 , f 2 are polynomials in 1 , . . . , n with coefficients in R1. Now
q Ž a j f 1 Ž . a ⬙ . s q Ž f 1 Ž . a ⬙ a j . s q Ž f 1 Ž . a ⬘ . . But we have ⬘Ž i . s j s Ž i . q 1, and also ⬙ Ž i . s j. Hence the induction hypothesis works. Next assume that l Ž ⬘. - l Ž ⬙ .. Then a ⬙ s a ⬘ a j and a s a j a ⬘ a j . Hence
q Ž a w . s q Ž 1c1 ⭈⭈⭈ nc n a j a ⬘ a j . s q Ž a j f 1 Ž . a ⬘ a j . q q Ž f 2 Ž . a ⬘ a j . . But since
q Ž a j f 1 Ž . a ⬘ a j . s q Ž f 1 Ž . a ⬘ a2j . s Ž q y qy1 . q Ž f 1 Ž . a ⬙ . q q Ž f 1 Ž . a ⬘ . ,
852
TOSHIAKI SHOJI
we have
q Ž a w . s q Ž f 1 Ž . a ⬘ . q q Ž f 3 Ž . a ⬙ . for certain polynomials f 1 , f 3 . Since ⬘Ž i . s ⬙ Ž i . s j s Ž i . q 1, the induction hypothesis works. The latter assertion is easily checked for both cases. Thus we get Ž7.5.1.. We now prove the proposition by induction on l 1Ž w .. By Ž7.5.1., we may assume that w g ˆ ⌺. Then w is a product of the elements of the form c ly 1 c l t ly1 t l ⭈⭈⭈ t kc k sk sky1 ⭈⭈⭈ sl for some integers l F k. Accordingly, a w is a c ly 1 c l product of Žmutually commuting. elements of the form ly1 l ⭈⭈⭈ ck k a k a ky1 ⭈⭈⭈ a l . We consider the special case where w s t 1c1 t 2c 2 ⭈⭈⭈ t nc n sn sny1 ⭈⭈⭈ s2 . Put s sn ⭈⭈⭈ s2 . Then c ny 2 c n c ny1 q Ž a w . s q Ž 1c1 ⭈⭈⭈ ny2 n a n a ny1 ⭈⭈⭈ a2 ny1 . c ny 2 qc ny 1 c n s q Ž 1c1 ⭈⭈⭈ ny2 n a n a ny1 ⭈⭈⭈ a2 . q
Ý
⬘-
q Ž f ⬘ Ž . a ⬘ .
by Ž3.6.5. ᎐ Ž3.6.8., with some polynomials f ⬘ on . Then by repeating this procedure for ny 2 , ny3 , . . . , 1 , we have
q Ž a w . s q Ž nc1q ⭈⭈⭈ qc n a n a ny1 ⭈⭈⭈ a2 . q
Ý
⬘-
q Ž fX ⬘ Ž . a ⬘ .
fX ⬘
for some polynomials on . Now z s nc1q ⭈⭈⭈ qc n a n ⭈⭈⭈ a2 g ⌺ for some g Pn, r . On the other hand, Proposition 6.10 and Theorem 6.14 imply Žby the linear independence of Schur functions. that q Ž a z . s q Ž a .. Hence the proposition holds in this case. The argument for the general case is completely similar. Remark 7.6. Proposition 7.5 and Corollary 6.15 imply that q Ž a w . g R1. But this also follows from Proposition 4.3. More precisely, we see that q Ž a w . is mapped to 1 Ž w . by the specialization 0 . In fact, by Proposition 4.3, any irreducible representation Z has an R1 form Žsince an irreducible Hn-module is realized on R1 .. Moreover, by the specialization 0 this turns out to be an irreducible W-module Z . The above assertions follow easily from this. 7.7. Let Wqs Wq Ž . be as in 5.1, and let 1 be the linear character of Wq as given in 5.5. We consider the induced character IndW W q 1 of W, which will be denoted by ⌰ 1 . The value of ⌰ 1 at w is interpreted in terms of PŽ x . and m Ž x . as follows Žrecall that PŽ x . is the complex conjugate of PŽ x ..: P Ž x . s
Ý
g Pn, r
⌰1 Ž w . m Ž x . .
Ž 7.7.1.
ARIKI ᎐ KOIKE ALGEBRAS
853
In fact, if we write P s Ý c , m , the coefficient c , is described as follows. Let G be the set of maps from the set X s Ž i, j . ¬ 1 F i F r, 1 F j F l Ž Ž i. .4 to 1, 2, . . . , r 4 . Then g g G determines a multipartition Ž g . s Ž Ž1., . . . , Ž r . . g Pn, r , where Ž k . is obtained from the set Žj i. ¬ g Ž i, j . s k 4 by arranging the order. Let F Ž g, i, . be the set of maps f from the set 1, 2, . . . , l Ž Ž i. .4 to 1, 2, . . . , r 4 such that Ý j jŽ i. s Žki. Žwhere the sum is taken over all j such that f Ž j . s k .. Then, it follows from the definition of P and m that we can write r
c , s
Ý
Ł
gg G Ž i , j .gX
i⭈ g Ž i , j. Ł F Ž g , k , . .
Ž 7.7.2.
ks1
On the other hand, it is easy to compute the value ⌰ 1 Ž w . explicitly, which coincides with the value c , in Ž7.7.2.. Thus Ž7.7.1. holds. For each partition s Ž 1 , 2 , . . . . of n, we set zŽ t . s z ⭈ Ł i G 1Ž1 y i .y1 t , where z is the order of the centralizer of in ᑭ n . For a multipartition s Ž Ž1., . . . , Ž r . ., we define a function z Ž t . by r
z Ž t . s r lŽ . Ł zŽ i. Ž t . .
Ž 7.7.3.
is1
For a partition s Ž 1 , 2 , . . . , k ., the Hall᎐Littlewood function k qŽ x; t . is defined as qŽ x; t . s Ł is1 q iŽ x; t .. For a multipartition g Pn, r , we define a function q Ž x; t . by r
qŽ x ; t . s
Ł q Ž x Ž i. ; t . .
Ž 7.7.4.
Ž i.
is1
We now introduce infinitely many variables, x iŽ k ., yiŽ k ., for i s 1, 2, . . . , and 1 F k F r. We have the following lemma. LEMMA 7.8.
Let r
⍀ Ž x, y ; t . s
ŁŁ
ks1 i , j
1 y tx iŽ k . y jŽ k . 1 y x iŽ k . y jŽ k .
.
Then we ha¨ e ⍀ Ž x, y ; t . s
Ý qŽ x ; t . m Ž y . ,
Ž 7.8.1.
Ý z Ž t . y1 P Ž x . P Ž y . ,
Ž 7.8.2.
⍀ Ž x, y ; t . s
where in both formulas, runs o¨ er the multipartitions s Ž Ž1., . . . , Ž r . . of any size.
854
TOSHIAKI SHOJI
Proof. The equality Ž7.8.1. follows easily from the equality for the case where r s 1, Žcf. wM, III, Ž4.2.x.. The equality Ž7.8.2. for r s 1 is given in wM, III, Ž4.1.x. By a similar argument as in the case r s 1, we have log ⍀ Ž x, y ; t . s
⬁
1 y tm
ÝÝ Ý k
m
i , j ms1
Ž x iŽ k . yjŽ k . .
m
.
But since r
1
Ý Ž yk k ⬘ .
r
as1
r
⬁
a
s ␦k , k⬘ ,
we have log ⍀ Ž x, y ; t . s
Ý Ý
1 y tm
Ý
as1 ms1 k , k ⬘, i , j ⬁
r
s
Ý Ý as1 ms1
1 y tm rm
rm
yak ak ⬘ Ž x iŽ k . y jŽ k ⬘. .
m
PmŽ a. Ž x . PmŽ a. Ž y . .
Hence r
⍀ Ž x, y ; t . s
⬁
Ł Ł exp
as1 ms1
s
½
1 y tm rm
PmŽ a. Ž x . PmŽ a. Ž y .
5
Ý z Ž t . y1 P Ž x . P Ž y . ,
as asserted. Thus, the lemma is proved. 7.9. Returning to the Hecke algebra setting, we shall express the character value q Ž a . in terms of the characters 1 and ⌰1 Žfor some . of W. For this, we introduce some notation. For a partition s Ž 1 , 2 , . . . , k ., we denote by M the set of k = r matrices C s Ž c i j . with integers c i j G 0 such that Ý rjs1 c i j s i . For C g k M, we define an element u C s Ł is1 u nŽ i. , where nŽ i . is the largest j such that c i j / 0 Žfor a fixed i .. Also we associate s Ž C . s Ž Ž1., . . . , Ž r . . g Pn, r to C g M as follows: the partition Ž j. is obtained from the set c i j ¬ 1 F i F k 4 by arranging the order. More generally, for a multipartition g Pn, r , we consider the set M s MŽ1. = ⭈⭈⭈ = MŽ r . , and for each C s Ž C1 , . . . , Cr . g M , we put u C s r Ł is1 u Ci i . The multipartition s ŽC. s Ž Ž1., . . . , Ž r . . associated to C g M is defined as the union of Ž Ci .; i.e., Ž j. is the union of corresponding Ž jth. partitions in Ž Ci ..
ARIKI ᎐ KOIKE ALGEBRAS
855
Now by making use of Proposition 6.12, q˜Ž x; q, t . may be interpreted by the following formula: q˜ Ž x ; q, u . s
qn
Ž q y qy1 .
lŽ .
Ý
u C q ŽC. Ž x ; qy2 . .
Ž 7.9.1.
Cg M
We have the following result. THEOREM 7.10.
q Ž a . s
For each , g Pn, r , we ha¨ e qn y1 l Ž .
Žq y q .
Ý
uC
Ý
g Pn , r
Cg M
z Ž qy2 .
y1
1 Ž w . ⌰1 ŽC. Ž w . .
Proof. The proof is similar to wR, Theorem 5.4x. First we compute the coefficient b , of S Ž x . in the expansion of q Ž x; t . in terms of Schur functions. Then b , coincides with the coefficient of S Ž x . m Ž y . in ⍀ Ž x, y; t . by Ž7.8.1.. Hence by Ž7.8.2., it is equal to the coefficient of S Ž x . m Ž y . in Ý z Ž t .y1 P Ž x . P Ž y .. But Ž6.1.4. implies that P Ž x . s
Ý 1 Ž w . S Ž x . .
It follows, by virtue of Ž7.7.1., that b , s
Ý z Ž t . y1 1 Ž w . ⌰1 Ž w . .
Ž 7.10.1.
Now by the Frobenius formula ŽTheorem 6.14., q Ž a . coincides with the coefficient of S Ž x . in q˜Ž x; q, u.. Hence the theorem follows from Ž7.9.1. together with Ž7.10.1.. Remark 7.11. Theorem 7.10 expresses the character value q Ž a . in terms of the values of irreducible characters and certain induced characters of W. The corresponding formula for Hn was given in wR, Theorem 5.4x. Our proof of Theorem 7.10 is a generalization of his argument. However, essentially the same result as Ram had already been obtained by Starkey wSx in his Ph.D. thesis in 1975, though it was not published at all. The author is grateful to M. Geck for communicating him the result of Starkey. See wGex for more details on Starkey’s work and related topics.
REFERENCES wAKx S. Ariki and K. Koike, A Hecke algebra of Ž⺪rr⺪. X ᑭ n and construction of its irreducible representations, Ad¨ . Math. 106 Ž1994., 216᎐243.
856
TOSHIAKI SHOJI
wBMx K. Bremke and G. Malle, Reduced words and a length function for GŽ e, 1, n., Indag. Math. 8 Ž1997., 453᎐469. wGex M. Geck, The character table of the Iwahori᎐Hecke algebra of the symmetric group: Starkey’s rule, C. R. Acad. Sci. Paris Ser. ´ I. Math. 329 Ž1999., 361᎐366. wGPx M. Geck and G. Pfeiffer, On the irreducible characters of Hecke algebras, Ad¨ . Math. 102 Ž1993., 79᎐94. wGx A. Gyoja, A q-analogue of Young symmetrizer, Osaka J. Math. 23 Ž1986., 841᎐852. wJx M. Jimbo, A q-analogue of UŽ ᒄ ᒉ Ž N q 1.., Hecke algebra, and the Yang᎐Baxter equation, Lett. Math. Phys. 11 Ž1986., 247᎐252. wMx I. G. Macdonald, ‘‘Symmetric Functions and Hall Polynomials,’’ 2nd ed., Clarendon, Oxford, 1995. wPx G. Pfeiffer, Character values of Iwahori᎐Hecke algebras of type B, in ‘‘Finite Reductive Groups, Related Structures and Representations ŽM. Cabanese, Ed.., Progress in Math. Vol. 141, pp. 333᎐360, Birkhauser, Boston, 1997. ¨ wRx A. Ram, A Frobenius formula for the characters of the Hecke algebras, In¨ ent. Math. 106 Ž1991., 461᎐488. wRSx K. Rampetas and T. Shoji, Length functions and Demazure operators for GŽ e, 1, n., I, Indag. Math. 9 Ž1998., 563᎐580. wSSx M. Sakamoto and T. Shoji, Schur᎐Weyl reciprocity for Ariki᎐Koike algebras, J. Algebra 221 Ž1999., 293᎐314. wSx A. J. Starkey, ‘‘Characters of the Generic Hecke Algebra of a System of BN-pairs,’’ Ph.D. thesis, University of Warwick, 1975. wWx H. Wenz, Hecke algebras of type A n and subfactors, In¨ ent. Math. 92 Ž1988., 349᎐383.
A Frobenius Formula for the Characters of Ariki᎐Koike Algebras1 Toshiaki Shoji Department of Mathematics, Science Uni¨ ersity of Tokyo, Noda, Chiba 278-8510, Japan Communicated by Michel Broue´ Received June 10, 1999
Let Hn, r be the Ariki᎐Koike algebra associated to the complex reflection group Wn, r s GŽ r, 1, n.. In this paper, we give a new presentation of Hn, r by making use of the Schur᎐Weyl reciprocity for Hn, r established by M. Sakamoto and T. Shoji Ž1999, J. Algebra, 221, 293᎐314.. This allows us to construct various non-parabolic subalgebras of Hn, r . We construct all the irreducible representations of Hn, r as induced modules from such subalgebras. We show the existence of a partition of unity in Hn, r , which is specialized to a partition of unity in the group algebra ⺓Wn, r . Then we prove a Frobenius formula for the characters of Hn, r , which is an analogy of the Frobenius formula proved by A. Ram Ž1991, In¨ ent. Math. 106, 461᎐488. for the Iwahori᎐Hecke algebra of type A. 䊚 2000 Academic Press
Contents 1. Introduction 2. Cyclotomic Schur᎐Weyl reciprocity 3. A New Presentation of HK 4. Irreducible Representations of HK 5. Orthogonal Primiti¨ e Idempotents 6. A Frobenius Formula for the Characters of HK 7. Character Values
1. INTRODUCTION In wRx, Ram gave a Frobenius type formula for the characters of the Hecke algebra Hn associated to the symmetric group ᑭ n . He derived 1
This paper is a contribution to the Joint Research Project ‘‘Representation Theory of Finite and Algebraic Groups’’ 1997᎐1999 under the Japanese᎐German Cooperative Science Promotion Program supported by JSPS and DFG. 818 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
ARIKI ᎐ KOIKE ALGEBRAS
819
the formula by making use of the Schur᎐Weyl reciprocity between Hn and the quantum group Uq Ž gl m ., established by Jimbo wJx. Let Hn, r be the Ariki᎐Koike algebra, i.e., the cyclotomic Hecke algebra associated to the complex reflection group Wn, r s GŽ r, 1, n.. The Schur᎐Weyl reciprocity was generalized in wSSx to the case between Hn, r and a certain Levi subalgebra of Uq Ž gl m .. In this paper, we give a Frobenius type formula for the characters of Hn, r , based on the Schur᎐Weyl reciprocity mentioned above. ŽNote that in this introduction, the algebra Hn, r stands for HK in Section 2.. One of the crucial ingredients in wRx is the existence of a partition of unity in Hn , i.e., a complete system of primitive orthogonal idempotents, which is specialized to a partition of unity in the group algebra ⺓ ᑭ n . This result had been proved independently by Gyoja wGx and Wenzl wWx. We show that a similar fact also holds for Hn, r . ŽIn the first draft of the paper, this result was used to obtain the Frobenius formula for HK . However, as pointed out by the referee, there exists an alternative argument for it without relying on this fact. Nevertheless, we left the part discussing a partition of unity unchanged, as we believe that it is of independent interest.. First we give a new presentation of Hn, r based on the result in wSSx. We show that Hn, r is generated by 1 , . . . , n and a2 , . . . , a n , where a2 , . . . , a n are generators of Hn as usual, and 1 , . . . , n satisfy a common relation of degree r. Moreover, 1 , . . . , n generate an abelian subalgebra of Hn, r . Note that our abelian subalgebra is different from the one defined in wAKx. The relationship between our generators and generators a1 , . . . , a n given in wAKx is complicated. But since our generators j are symmetric for j s 1, . . . , n, it makes it possible to embed the algebra Hn1 , r m Hn 2 , r m ⭈⭈⭈ into Hn, r , whenever Ýn i F n, just as in the group case. ŽPfeiffer wPx considered certain non-parabolic subalgebras in the case of Iwahori᎐Hecke algebras of type B based on the standard generators. However, its structure is more complicated than ours.. Then we can construct all the irreducible representations of Hn, r as induced modules from certain Žnon-parabolic. subalgebras, in analogy to the group case construction. Under this setting, one can generalize the method of Gyoja in the case of type A to our case. A Frobenius formula is given as follows. We define some elements a in Hn, r , parametrized by multipartitions s Ž Ž1., . . . , Ž r . . of size n. Note that they are mapped, by the specialization to ⺓Wn, r , to a complete system of representatives of conjugacy classes in Wn, r . Let q be an irreducible character of Hn, r associated to a multipartition of size n. Then we have q˜ Ž x ; q, u . s
Ý q Ž a . S Ž x . ,
820
TOSHIAKI SHOJI
where q˜Ž x; q, u. is a certain polynomial, which can be described by using Hall᎐Littlewood symmetric functions qnŽ x; t ., and S Ž x . is the Schur function associated to a multipartition Žsee Section 6 for the detailed definition.. As an application one can show, for any character q of Hn, r , that the character values q Ž a w . are determined by a simple algorithm from the values q Ž a ., where a w ¬ w g Wn, r 4 is a basis of Hn, r defined in terms of our generators. The author is grateful to A. Ram for stimulating discussions on the occasion of a symposium in Kyoto, 1998. He also thanks the referee for his suggestions on a partition of unity of HK .
2. CYCLOTOMIC SCHUR᎐WEYL RECIPROCITY 2.1. Let R s ⺪w q, qy1 , u1 , . . . , u r x be a polynomial ring over ⺪ with indeterminates q, u1 , . . . , u r . The Ariki᎐Koike algebra, i.e., the cyclotomic Hecke algebra associated to the complex reflection group Wn, r s GŽ r, 1, n., is the algebra H s Hn, r over R with generators a1 , . . . , a n subject to the following conditions:
Ž a1 y u1 . ⭈⭈⭈ Ž a1 y u r . s 0, Ž ai y q . Ž ai q qy1 . s 0
Ž i G 2. ,
a1 a2 a1 a2 s a2 a1 a2 a1 , ai a j s a j ai a i a iq1 a i s a iq1 a i a iq1
Ž < i y j < G 2. , Ž 2 F i F n y 1. .
Let K be the quotient field of R. We set W s Wn, r . It is known by wAKx that H is a free R-module with rank < W < s n!r n, and that HK s K mR H is a semisimple algebra over K. r 2.2. Let V s [is1 Vi be a free R-module with Vi s R m i . We set r m s Ý is1 m i . Let us fix a basis E s ¨ 1 , . . . , ¨ m 4 of V such that
Ei s ¨ j ¬ m1 q ⭈⭈⭈ qm iy1 - j F m1 q ⭈⭈⭈ qm i 4 gives rise to a basis of Vi . We define a map b: 1, 2, . . . , m4 ª ⺞ by bŽ j . s i whenever ¨ j g Vi . Let wt: V ª V be a linear operator defined by wt Ž ¨ j . s u bŽ j.¨ j . Let us define linear operators, T, , S on V m2 as follows.
ARIKI ᎐ KOIKE ALGEBRAS
821
For ¨ i , ¨ j g E ,
¡q¨ m ¨ i
~
if i s j,
j
T Ž¨i m ¨j . s ¨ j m ¨i
¢¨ m ¨ q Ž q y q j
if i ) j,
y1
i
. ¨i m ¨j
if i - j,
Ž¨i m ¨j . s ¨j m ¨i , SŽ¨i m ¨j . s
½
T Ž¨i m ¨j .
if b Ž i . s b Ž j . ,
Ž¨i m ¨j .
if b Ž i . / b Ž j . .
Using these operators, we define operators Ti , i , Si , j g End R V mn, Ž2 F i F n, 1 F j F n., by the following rule: iy2. nyi. Ti s idmŽ m T m idmŽ , V V iy2. nyi. i s idmŽ m m idmŽ , V V iy2. nyi. Si s idmŽ m S m idmŽ , V ¨ jy1. nyj. j s idmŽ m wt m idmŽ . V V
We now define an operator T1 on V mn by T1 s Ty1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S2 1 . 2
Ž 2.2.1.
Then it is shown in wSS, Theorem 3.2x that : a i ¬ Ti Ž1 F i F n. gives rise to a representation of H on V mn. 2.3. Let Uq Ž gl m . be the quantized universal enveloping algebra of gl m over K. Hence, Uq Ž gl m . is an associative algebra with generators, e i , f i , Ž1 F i F m y 1. and q " i , Ž1 F i F m., and well-known relations Žsee, e.g., wSSx.. We consider the natural representation of Uq Ž gl m . on VK s K mR V, which is given by
Ž ei . ¨ j s Ž fi . ¨ j s Ž q" i .¨j s
½ ½
½
¨ jy1
if j s i q 1,
0
if j / i q 1,
¨ jq1
if j s i ,
0
if j / i ,
q"1¨ j
if j s i ,
0
if j / i.
822
TOSHIAKI SHOJI
By the comultiplication ⌬: Uq Ž gl m . ª Uq Ž gl m . m Uq Ž gl m ., induces a representation of Uq Ž gl m . on VKmn, which we denote also by . It is expressed on generators as ny1
Ž ei . s
Ý
K im p m e i m 1mŽ ny1yp. ,
ps0 ny1
Ž fi . s
Ý
1m p m f i m Ž Ky1 i .
m Ž ny1yp .
,
ps0
Ž q " i . s q " i m ⭈⭈⭈ m q " i , where K i stands for q iy iq 1 for i s 1, . . . , m y 1. Let ᒄ s gl m 1 [ ⭈⭈⭈ [ gl m r be a Levi subalgebra of gl m and Uq Ž ᒄ . the subalgebra of Uq Ž gl m . associated to ᒄ. By the restriction, Uq Ž ᒄ . acts on VKmn, which is again denoted by . Let s Ž Ž1., . . . , Ž r . . be a multipartition consisting of r-tuples of partitions Ž i.. We denote by < < s Ý ris1 < Ž i. < the size of , where Ž i. s i Ž Ž1i., . . . , Žki. . is a partition of < Ž i. < s Ý kjs1 Žj i.. Let Pn, r be the set of i Ž1. Ž r . multipartitions s Ž , . . . , . such that < < s n. It is known by wAKx that all the irreducible representations of HK are Žup to isomorphism. parametrized by the set Pn, r . We denote by Z the irreducible HK-module corresponding to g Pn, r . On the other hand, irreducible representations of Uq Ž ᒄ . occurring in VKmn are parametrized by the set ⌳ m 1 , . . . , m r of multipartitions g Pn, r such that l Ž Ž i. . F m i . ŽHere for a partition s Ž 1 , . . . , k ., we put l Ž . s k.. We denote by V the irreducible Uq Ž ᒄ .-module corresponding to g ⌳ m 1 , . . . , m r. Under this setting, the following Schur᎐Weyl reciprocity holds between Uq Ž ᒄ . and HK . THEOREM 2.4 wSS, Theorem 4.1x. Ž HK . and ŽUq Ž ᒄ .. are mutually the full centralizers of each other; i.e., we ha¨ e
Ž HK . s End Uq Ž ᒄ . VKmn ,
Ž Uq Ž ᒄ . . s End HK VKmn .
Moreo¨ er, the HK m Uq Ž ᒄ .-module VKmn is decomposed as VKmn ,
[
g⌳ m1 , . . . , m r
Z m V .
ARIKI ᎐ KOIKE ALGEBRAS
823
One can show that Si and j commute with the action of Uq Ž ᒄ . Žcf. wSS, Lemma 3.4x.. Hence as a corollary to Theorem 2.4, we have COROLLARY 2.5. The operators Si Ž2 F i F n., j Ž1 F j F n. are contained in Ž HK .. 2.6. We fix the set of standard generators t 1 , s2 , . . . , sn4 of W, where t 1r s 1 and s2 , . . . , sn are standard generators of the symmetric group ᑭ n under the isomorphism W , ᑭ n h Ž⺪rr⺪. n. We put t i s si ⭈⭈⭈ s2 t 1 s2 ⭈⭈⭈ si for i s 2, . . . , n. Then t 1 , . . . , t n are generators of Ž⺪rr⺪. n, and any element w g W can be written in a unique way as w s t 1c1 ⭈⭈⭈ t nc n , where g Sn , and c i are integers such that 0 F c i F r y 1. Let V s [Vi with Vi s ⺓ m i . We fix a basis E s ¨ 1 , . . . , ¨ m 4 of V in a similar way as in 2.2. We fix hereafter a primitive r th root of unity in ⺓. The group W acts on the n-fold tensor space V mn ; each si acts as i as in 2.2, and t j acts as jy1. nyj. idmŽ m wt m idmŽ , where wt is a linear map on V defined by wtŽ ¨ j . V V bŽ j. s ¨ j . We denote this representation of W by . Let 0 : R ª ⺓ be the specialization homomorphism defined by 0 Ž q . s 1, 0 Ž u i . s i. Let ⺓W be the group algebra of W over ⺓. By the specialization 0 , one obtains ⺓ mR H , ⺓W. Moreover, via the specialization ⺓ mR V mn , V mn, 1 m Ti and 1 m Si coincide with Ž si . on V mn, and 1 m j coincides with Ž t j .. Note that it is known by wAKx that the subalgebra of H generated by a2 , . . . , a n4 coincides with the Hecke algebra Hn associated to ᑭ n . Then one can define an element a for any g ᑭ n as the product a s a i1 ⭈⭈⭈ a i r according to the reduced expression s si1 ⭈⭈⭈ si r. We denote by T , for g S n , the operator Ž a . on V mn. Now assume that m i G n for i s 1, . . . , n. Then it is known wSS, Proposition 5.3x that the representations and are faithful. In particular, in this case, Ž w . ¬ w g W 4 are linearly independent operators on V mn. For each w s t 1c1 ⭈⭈⭈ t nc n g W, we define an operator Tw on V mn by Tw s 1c1 2c 2 ⭈⭈⭈ nc n T . Then Tw is mapped, by the specialization 0 , to Ž t 1c1 ⭈⭈⭈ t nc n . s Ž w .. Hence by using a similar argument as in the proof of Proposition 5.3 in wSSx, we see that Tw ¬ w g W 4 are linearly independent over R. Since dim K Ž HK . s < W <, we have the following. PROPOSITION 2.7. Assume that m i G n for 1 F i F r. Then Tw ¬ w g W 4 gi¨ es rise to a basis of Ž HK .. 3. A NEW PRESENTATION OF HK 3.1. In this section, we assume that m i G n for each i. In the remainder of this paper, we use the following notation. Let M be an R-module. For any commutative algebra R⬘ over R, we write MR⬘ s R⬘ mR M.
824
TOSHIAKI SHOJI
Let V s [Vi as in 2.2. We write the basis Ei of Vi as ¨ jŽ i. ¬ 1 F j F m i 4 , by preserving the orders. Let U be the R-submodule of V mn spanned by 1 . m ⭈⭈⭈ m ¨ Ž c n . , with g ᑭ ¨Ž cŽ1. Ž n. n and 1 F c i F r for i s 1, . . . , n. Then U is a free R-module of rank < Wn, r <. Note that U is H-stable. In fact, it is clear that U is stable by Ti and Si for i s 2, . . . , n. Since U is stable by 1 , U is also stable by T1. Hence it is stable by H . Let us define an element ¨ 0 g U by r
¨0 s
ž
r
Ý ¨ nŽ i . 1
i1s1
m
/ ž
r
Ži . Ý ¨ ny1 2
i 2s1
/
m ⭈⭈⭈ m
ž
Ý
i n s1
/
¨ 1Ž i1 . .
We have the following lemma. LEMMA 3.2. The HK-module UK is generated by ¨ 0 . In particular, UK is isomorphic to the regular HK-module. Proof. We consider the W-module V mn. We write the basis Ei of Vi as ¨ jŽ i. ¬ 1 F j F m i 4 as above, and define a subspace U of V mn in a similar way as above by replacing ¨ jŽ i. by ¨ jŽ i.. Then U is a ⺓W-submodule of V mn. We also define an element ¨ 0 g U in a similar way. We claim that Ž3.2.1. The ⺓W-module U is generated by ¨ 0 . In order to show the claim, it is enough to see that ⺓W¨ 0 contains all Ž i. 1 . m ⭈⭈⭈ m ¨ Ž c n . . Put f s Ý r the vectors of the form ¨Ž cŽ1. Ž n. j is1 ¨ j . We consider a the action of t j on ¨ 0 s f n m ⭈⭈⭈ m f 1. It only affects the jth component of Ž i. ¨ 0 . The jth component of t ja ¨ 0 is equal to Ý ris1 ai ¨ nyjq1 for a s 0, . . . , r y 1. It follows that Ž i. f n m ⭈⭈⭈ m f nyjq2 m ¨ nyjq1 m f nyj m ⭈⭈⭈ m f 1 g ⺓W¨ 0 ,
for i s 1, . . . , r. Repeating this procedure for j s 1, . . . , n, we see that
¨ nŽ c1 . m ⭈⭈⭈ m ¨ 1Ž c n . is contained in ⺓W¨ 0 for any 1 F c i F r. Then by apply1 . m ⭈⭈⭈ m ¨ Ž c n . g ⺓W¨ . ing g ᑭ n on these vectors, we conclude that ¨Ž cŽ1. Ž n. 0
The claim follows. Now Ž3.2.1. implies that the set w¨ 0 ¬ w g Wn, r 4 is linearly independent over ⺓. Let Tw ¬ w g W 4 be the basis of HK given in 2.6. Then by the specialization 0 , Tw¨ 0 is mapped to w¨ 0 for each w g W. Hence we see that the set Tw¨ 0 ¬ w g W 4 is linearly independent over R. In particular, dim K HK ¨ 0 s < W <. Since HK ¨ 0 ; UK , we conclude that HK ¨ 0 s UK . The lemma is proved. 3.3. We write ¨ 0 as ¨ 0 s f n m ⭈⭈⭈ m f 1 , where f j s Ý ris1 ¨ jŽ i.. The action of ja on ¨ 0 affects only the jth factor f nyjq1 , which is mapped to Ž i. Ý ris1 u ia ¨ nyjq1 . Let A be the matrix of degree r whose ab-entry is equal to
ARIKI ᎐ KOIKE ALGEBRAS
825
u ba for 1 F b F r, 0 F a F r y 1; i.e., A is the usual Vandermonde matrix. Thus ⌬ s det A is the Vandermonde determinant, ⌬ s Ł i ) j Ž u i y u j .. We can write the inverse of A as Ay1 s ⌬y1 B, where B s Ž h b aŽ u.. and h b aŽ u. is a polynomial in ⺪w u1 , . . . , u r x. Under this setting, we have Ž b. f n m ⭈⭈⭈ m f nyjq2 m ¨ nyjq1 m f nyj m ⭈⭈⭈ m f 1 s ⌬y1
ry1
Ý h b a Ž u . ja ⭈ ¨ 0 . as0
Ž 3.31. We now introduce a polynomial Fc Ž X . with a variable X, for 1 F c F r, with coefficients in ⺪wŽ u1 , . . . , u r x by Fc Ž X . s
Ý
h ci Ž u . X i .
Ž 3.3.2.
0FiFry1
The following formula is easily obtained from the definition: Fc Ž u c⬘ . s ⌬ ⭈ ␦cc⬘ .
Ž 3.3.3.
Take j G 2. By repeating the procedure for getting Ž3.3.1. to the action of a jy1 , we have Ž c1 . Ž c2 . f n m ⭈⭈⭈ m f nyjq3 m ¨ nyjq2 m ¨ nyjq1 m f nyj m ⭈⭈⭈ m f 1
s ⌬y2 Fc1Ž jy1 . Fc 2Ž j . ⭈ ¨ 0 .
Ž 3.3.4.
Now by applying Tj on both sides of Ž3.3.4., we have the formula Ž c2 . Ž c1 . f n m ⭈⭈⭈ m f nyjq3 m ¨ nyjq1 m ¨ nyjq2 m f nyj m ⭈⭈⭈ m f 1
¡ T F Ž ¢Ž T y Ž q y q y2
s
~⌬
j
c1
jy1
. Fc 2Ž j . ⭈ ¨ 0
y1
j
y2
. Te . ⌬
Fc1Ž jy1 . Fc 2Ž j . ⭈ ¨ 0
if c1 G c 2 , if c1 - c 2 ,
Ž 3.3.5. where Te stands for the identity operator Žcorresponding to the unit element e g W .. We have the following lemma. LEMMA 3.4.
For j s 2, . . . , n, we ha¨ e
Tj j s jy1Tj q ⌬y2 Tj jy1 s j Tj y ⌬y2 Tj k s k Tj
Ý Ž uc
c1-c 2
Ý Ž uc
c1-c 2
2
2
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . ,
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . ,
Ž k / j y 1, j . .
826
TOSHIAKI SHOJI
In the expression of the formulas, the sum is taken o¨ er all 1 F c1 , c 2 F r such that c1 - c2 . Proof. It is clear that if k / j y 1, j, then Tj k s k Tj . We consider the first and second equalities. By Lemma 3.2, the map HK ª UK , h ¬ hŽ ¨ 0 ., is injective. Hence to prove the equality in the lemma, we have only to evaluate both sides at ¨ 0 . First we compare the values jy1Tj Ž ¨ 0 . and Tj jy1Ž ¨ 0 .. They only affect the j y 1 and jth factors of ¨ 0 s f n m ⭈⭈⭈ m f 1. The j y 1 and jth factor of jy1Tj Ž ¨ 0 . is equal to
Ý
c1Gc 2
Ž c2 . Ž c1 . u c 2¨ nyjq1 m ¨ nyjq2
q
Žc . Žc . Žc . Žc . m ¨ nyjq2 q Ž q y qy1 . u c ¨ nyjq2 m ¨ nyjq1 Ý Ž u c ¨ nyjq1 ., 2
1
2
c1-c 2
1
2
1
and the j y 1 and jth factor of Tj jy1Ž ¨ 0 . is equal to
Ý
c1Gc 2
Ž c2 . Ž c1 . u c1¨ nyjq1 m ¨ nyjq2
q
Žc . Žc . Žc . Žc . m ¨ nyjq2 q Ž q y qy1 . u c ¨ nyjq2 m ¨ nyjq1 Ý Ž u c ¨ nyjq1 .. 2
1
1
c1-c 2
1
2
1
It follows that jy1Tj Ž ¨ 0 . y Tj jy1Ž ¨ 0 . coincides with f n m ⭈⭈⭈ m f nyjq3 m
½ÝŽ
Ž c2 . Ž c1 . u c 2 y u c1 . ¨ nyjq1 m ¨ nyjq2 m f nyj m ⭈⭈⭈ m f 1 .
c1 , c 2
5
A similar formula holds also for Tj and j . Now by using Ž3.3.5., we have
k Tj y Tj k s ⌬y2 q
Ý Ž uc
c1-c 2
2
½ÝŽ c1)c 2
u c 2 y u c1 . Tj Fc1Ž jy1 . Fc 2Ž j .
y u c1 . Ž Tj y Ž q y qy1 . Te . Fc1Ž jy1 . Fc 2Ž j . , Ž 3.4.1.
5
for k s j y 1, j, where s 1 Žresp. y1. if k s j y 1 Žresp. k s j .. Here we note that
j y jy1 s ⌬y2
Ý Ž uc
c1)c 2
2
y u c1 . Fc1Ž jy1 . Fc 2Ž j . y Fc 2Ž jy1 . Fc1Ž j . 4 .
Ž 3.4.2.
ARIKI ᎐ KOIKE ALGEBRAS
827
In fact, we replace the action of Tj on VKmn by j . Then the argument used to prove Ž3.4.1. still valid, and we get the formula, which is obtained from Ž3.4.1. by specializing q s 1 Žand remaining u i unchanged.. Then we have
k j y j k s ⌬y2j
Ý Ž uc
c1)c 2
2
y u c1 . Fc1Ž jy1 . Fc 2Ž j . y Fc 2Ž jy1 . Fc1Ž j . 4 .
Ž 3.4.3. But since jy1 j s j j , we have jy1 j y j jy1 s j Ž j y jy1 .. Hence Ž3.4.2. follows from Ž3.4.3.. Now by virtue of Ž3.4.2., the equality Ž3.4.1. can be rewritten
k Tj y Tj k s Tj Ž j y jy1 . y ⌬y2
Ý Ž uc
c1-c 2
2
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . .
The equalities in the lemma follow from this. Next we show LEMMA 3.5.
For j s 2, . . . , n, we ha¨ e
S j s Tj y ⌬y2 Ž q y qy1 .
Ý
c1-c 2
Fc1Ž jy1 . Fc 2Ž j . .
Proof. It is easily checked that for j s 2, . . . , n, we have S j Ž f n m ⭈⭈⭈ m f 1 . s j Ž f n m ⭈⭈⭈ m f 1 . .
Ž 3.5.1.
Then Ž3.3.5. implies that S j s ⌬y2
Ý
c1Gc 2
q ⌬y2
Tj Fc1Ž jy1 . Fc 2Ž j .
Ý Ž Tj y Ž q y qy1 . Te . Fc Ž jy1 . Fc Ž j . . Ž 3.5.2. 1
c1-c 2
2
As in the proof of Lemma 3.4, we replace Tj by j . Then S j is changed to j . By specializing q s 1 in the above formula, we have 1 s ⌬y2
Ý
Fc1Ž jy1 . Fc 2Ž j . .
c1 , c 2
Substituting Ž3.5.3. into Ž3.5.2., we obtain the required formula.
Ž 3.5.3.
828
TOSHIAKI SHOJI
3.6. Let R1 s ⺪w q, qy1 , u1 , . . . , u r , ⌬y1 x be the Ž⺪-. subalgebra of K generated by R and ⌬y1 . We define an algebra H h over R1 as an associative algebra generated by a2 , . . . , a n and 1 , . . . , n subject to the following relations:
Ž ai y q . Ž ai q qy1 . s 0
Ž 2 F i F n. ,
Ž 3.6.1.
Ž i y u1 . ⭈⭈⭈ Ž i y u r . s 0
Ž 1 F i F n. ,
Ž 3.6.2.
Ž 2 F i F n. ,
Ž 3.6.3.
ai a j s a j ai
Ž < i y j < G 2. ,
Ž 3.6.4.
i j s j i
Ž 1 F i , j F n . , Ž 3.6.5.
a i a iq1 a i s a iq1 a i a iq1
a j j s jy1 a j q ⌬y2
Ý Ž uc
c1-c 2
2
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . ,
Ž 3.6.6. a j jy1 s j a j y ⌬y2 aj k s k aj
Ý Ž uc
c1-c 2
2
y u c1 . Ž q y qy1 . Fc1Ž jy1 . Fc 2Ž j . , Ž 3.6.7.
Ž k / j y 1, j . .
Ž 3.6.8.
Then we have the following theorem. THEOREM 3.7. H h is a free R1-module with rank H h s < W <. Moreo¨ er, K mR 1 H h is isomorphic to HK , and HR 1 is isomorphic to a subalgebra of H h. Proof. For a reduced expression s si1 ⭈⭈⭈ si k g ᑭ n , the product a i1 ⭈⭈⭈ a i k is independent of the choice of the reduced expression Žthis follows from Ž3.6.1., Ž3.6.3., and Ž3.6.4.., which we denote by a . It follows from Ž3.6.6. ᎐ Ž3.6.8. that we see H h is generated over R1 by the elements of the form 1c1 ⭈⭈⭈ nc n a , with g ᑭ n and with integers c k such that 0 F c k F r y 1. We want to show that these elements are actually linearly independent over R1. Let us consider the tensor space VRmn . We note that the corre1 spondence a i ¬ Ti , j ¬ j gives rise to a representation of H h on VRmn . 1 In fact, j acts diagonally on the basis elements on VRmn with eigenvalues 1 u1 , . . . , u r 4 . Hence they satisfy the relations obtained by replacing j by j in Ž3.6.2.. By Lemma 3.4, the relations corresponding to Ž3.6.6. ᎐ Ž3.6.8. are also satisfied. Other relations are trivial. Now let be the thus obtained representation of H h on VRmn . Since j g Ž HK . by Corollary 2.5, we have 1 Ž HKh. ; Ž HK .. By Proposition 2.7, the elements 1c1 ⭈⭈⭈ nc n T 4 are linearly independent over K, and so are linearly independent over R1. This implies that 1c1 ⭈⭈⭈ nc n a 4 are also linearly independent over R1 as asserted.
ARIKI ᎐ KOIKE ALGEBRAS
829
The above proof also shows that the representation is faithful. Now we have dim K Ž HK . s < Wn, r <, and is faithful. It follows that HKh , HKh s Ž HK . , HK .
ž /
Next we note, by Lemma 3.5, that S j belongs to Ž H h.. It follows that T1 is contained in Ž H h., and we have Ž HR 1 . ; Ž H h.. Since and are faithful, this implies the last statement. The theorem is now proved. Remark 3.8. Theorem 3.7 asserts that there exist embeddings H ¨ H h ¨ HK . A priori, these embeddings depend on the choice of the representation , i.e., on the space V s [Vi . However, the embeddings are independent of the choice of . In fact, we define an element L i g Ž H . by L1 s T1 and by L i s Ti L iy1Ti inductively for i s 2, . . . , n. Then Lc11 ⭈⭈⭈ Lcnn T 4 gives rise to an R-basis of Ž H . by wAKx. Now by Lemmas 3.4 and 3.5, each Lc11 ⭈⭈⭈ Lcnn T can be written as a linear combination of elements in 1c1 ⭈⭈⭈ nc n T 4 , with coefficients independent of the choice of . It follows that each j is also expressed as a linear combination of elements in Lc11 ⭈⭈⭈ Lcnn T 4 , with coefficients independent of . This shows that the embeddings do not depend on the choice of . By making use of Theorem 3.7, we have the following refinement of Lemma 3.2. COROLLARY 3.9. Let UR 1 s R1 mR U . Then we ha¨ e UR 1 s H h¨ 0 . In particular, UR 1 is isomorphic to H h as left H h-modules. Proof. It is clear that H h¨ 0 is contained in UR 1. On the other hand, the arguments in 3.3 imply that ¨ nŽ c1 . m ⭈⭈⭈ m ¨ 1Ž c n . can be written as ¨ nŽ c1 . m ⭈⭈⭈ m ¨ 1Ž c n . s ⌬yn Fc1Ž 1 . Fc 2Ž 2 . ⭈⭈⭈ Fc nŽ n . ⭈ ¨ 0 .
This implies that UR 1 ; H h¨ 0 , and so we get the required equality, UR 1 s H h¨ 0 . Now the map H h ª UR 1, h ¬ h¨ 0 is surjective. Since it is injective by Lemma 3.2, it gives an isomorphism H h , UR 1.
3.10. We consider an example of H h in the case where n s r s 2. Hence H h is generated by a2 and 1 , 2 . In this case, ⌬ s u 2 y u1 , and we have F1Ž X . s u 2 y X, F2 Ž X . s yu1 q X. Then the commutation formula is given explicitly as a2 2 s 1 a2 y ⌬y1 Ž q y qy1 . Ž 1 y u 2 . Ž 2 y u1 . , a2 1 s 2 a2 q ⌬y1 Ž q y qy1 . Ž 1 y u 2 . Ž 2 y u1 . .
830
TOSHIAKI SHOJI
Put C s ⌬y2 Ž q y qy1 .. The operator S2 Žor rather the corresponding element in H h. can be computed by Lemma 3.5: S 2 s a 2 q C Ž 1 y u 2 . Ž 2 y u1 . . Now HK is the Hecke algebra of type B2 , and its basis elements are 1, a1 , a2 a1 , a1 a2 a1 4 and those multiplied by a2 to them from the right. They are expressed as a linear combination of elements of the form 1c1 2c 2 a as follows: a1 s 1 q Cu1 Ž 2 y u 2 . Ž 1 y u1 . a2 , a2 a1 s 2 a2 q Cu 2 Ž 2 y u1 . Ž 1 y u 2 . , a1 a2 a1 s 1 2 a2 q Cu1 u 2 Ž 2 y u1 . Ž 1 y u 2 . q Ž 2 y u 2 . Ž 1 y u1 . 4 a22 . Other basis elements are easily obtained from them. It seems interesting to find an explicit description of 1 , . . . , n in terms of a1 , a2 , . . . , a n . 4. IRREDUCIBLE REPRESENTATIONS OF HK 4.1. In wAKx, Ariki and Koike constructed an irreducible representation of HK corresponding to g Pn, r on the space spanned by the standard tableaux of shape . ŽFor the definition of the standard tableau of shape , see the next section.. In this section we shall give an alternate construction of irreducible representations of HK , which is more like the standard construction of irreducible representations of W in terms of induced representations from certain subgroups. 4.2. For s Ž Ž1., . . . , Ž r . . g Pn, r , we set < Ž i. < s n i , and pi s Ýiks1 n k . Let ᑭ Žni. be the subgroup of ᑭ n generated by s j such that piy1 q 1 - j F pi . Then ᑭ Žni. , ᑭ n i , and we define a parabolic subgroup ᑭ nŽ . of ᑭ n by Ž2. Žr. ᑭ n Ž . s ᑭ Ž1. n = ᑭ n = ⭈⭈⭈ = ᑭ n .
Let W Ž . be the subgroup of W obtained as the semidirect product of ᑭ nŽ . with Ž⺪rr⺪. n. Then W Ž . can be written as W Ž . s W Ž1. = W Ž2. = ⭈⭈⭈ = W Ž r . , where W Ž i. is the subgroup of W generated by ᑭ Žni. and t j such that piy1 - j F pi . We have W Ž i. , Wn i , r . Let Hn be the Hecke algebra over R associated to ᑭ n as given in 2.6. We denote by HnŽ i. the subalgebra of Hn generated by a j with piy1 q 1 - j
ARIKI ᎐ KOIKE ALGEBRAS
831
F pi . Thus HnŽ i. is isomorphic to the Hecke algebra Hn i . One can define a Žparabolic. subalgebra HnŽ . of Hn by Hn Ž . s HnŽ1. m HnŽ2. m ⭈⭈⭈ m HnŽ r . . We define an R1-algebra H Ž . as the subalgebra of H h generated by HnŽ . and 1 , . . . , n . Then it follows from 3.6 that H Ž . can be written as H Ž . s H
Ž1.
mH
Ž2.
m ⭈⭈⭈ m H
Žr.
,
where H Ž i. is the subalgebra of H h generated by HnŽ i. and j such that piy1 - j F pi . We note that H Ž i. , Ž Hn i , r . h. Set Hn,Ž i.K s K m HnŽ i.. We denote by ZŽ i. an irreducible Hn,Ž i.K-module corresponding to a partition Ž i. of n i under the identification HnŽ i. , Hn i . Then ZŽ i. can be extended to an irreducible HKŽ i.-module, where j g HKŽ i. acts on ZŽ i. as the scalar multiplication by u i . To see this, it is enough to show that the relations Ž3.6.1. ᎐ Ž3.6.8. hold once the j are replaced by the scalar multiplication u i , for piy1 - j F pi . The relations are trivial except the case Ž3.6.6. or Ž3.6.7.. Since Fc1Ž u i . Fc 2Ž u i . s 0 for c1 / c 2 by Ž3.3.3., the right hand side of Ž3.6.6. is equal to u i a j when j and jy1 are replaced by u i . This shows that Ž3.6.6. also holds. The verification of Ž3.6.7. is similar. We can now define an H Ž .K-module Z 0 as Z 0 s ZŽ1. m ZŽ2. m ⭈⭈⭈ m ZŽ r . .
Ž 4.2.1.
We obtain an induced module HK mH Ž . K Z0 . Then we have the following theorem. THEOREM 4.3. For each g Pn, r , HK mH Ž . K Z 0 is isomorphic to the irreducible HK-module Z . Proof. We shall construct HK mH Ž . K Z 0 as an HK-submodule of VKmn. i By the Schur᎐Weyl reciprocity between Hn i , K and Uq Ž gl m i . on Vi,mn K , Ž i. mn i Hn, K-module ZŽ i. can be realized as the subspace Hn i , K ¨Ž i. of Vi, K , where ¨Ž i. is a highest weight vector for an irreducible Uq Ž gl m i .-module corresponding to Ž i.. Then HnŽ .K-module Z 0 can be realized inside of VKmn, and it is written as Z 0 s HnŽ .K ¨ , where ¨ s ¨Ž1. m ¨Ž2. m ⭈⭈⭈ m ¨Ž r . g Z 0
is a highest weight vector in VKmn for an irreducible Uq Ž ᒄ .-module corresponding to g ⌳ m 1 , . . . , m r. Now the action of j on V mn is given by j . Then it is easy to see that j acts on Z 0 as a scalar multiplication by u i for piy1 - j F pi . Hence Z 0 can be extended to an H Ž .K-module which coincides with the one given in Ž4.2.1.. Let Zˆ be the HK-submodule of VKmn generated by Z 0 . Then Zˆ can be expressed as Zˆ s HK ¨ , and by
832
TOSHIAKI SHOJI
the Schur᎐Weyl reciprocity ŽTheorem 2.4., Zˆ is isomorphic to Z . On the other hand, by using the defining relations in 3.6 for H h, we see that HK is a free right H Ž .K-module with basis a w ¬ w g ᑭ nrᑭ nŽ .4 . But dim K Zˆ s dim K Z is equal to the dimension of the irreducible W-module corresponding to , which is given by r
ᑭ nrᑭ n Ž .
Ł dim K ZŽ i. s
is1
ᑭ nrᑭ n Ž . dim K Z 0 .
This implies that Zˆ , HK mH Ž . K Z 0 , and the theorem follows. 5. ORTHOGONAL PRIMITIVE IDEMPOTENTS 5.1. In wGx, Gyoja constructed a partition of unity, i.e., a complete system of orthogonal primitive idempotents in Hn, K which is specialized to a partition of unity in ⺓ ᑭ n . In this section, we generalize his result to the case of HK . A multipartition may be regarded as an r-tuple of Young diagrams. An r-tuple ⺣ s Ž S Ž1., . . . , S Ž r . . of tableaux is called a standard tableau of shape if each j Ž1 F j F n. occurs exactly once as an entry in one tableau in ⺣, and each S Ž i. is a standard tableau of shape Ž i.. For a g Pn, r , we define an r-tuple of standard tableau ⺣qs ⺣q Ž . as follows: we insert the numbers 1, 2, . . . , n in the r-tuple of Young diagrams by the following rule. First, insert the numbers 1, 2, . . . , n1 in the Young diagram of Ž1. in order along the consecutive rows in Ž1., from left to right, and from top to bottom, and then insert the numbers n1 q 1, . . . , n1 q n 2 in the Young diagram of Ž2. in the same way, and so on, until the last partition Ž r .. Thus ⺣q is defined. Next we define ⺣ys ⺣y Ž . in a similar way as ⺣q, but by using columns instead of rows in the Young diagrams Ž1., . . . , Ž r .. Ž1. Ž r .. We write ⺣ "s Ž S " , . . . , S" . Let Ž i. s Ž Ž1i., Ž2i., . . . . be a partition of Ž i. n1. Then Sq determines an embedding of ᑭ Ž1i. = ᑭ Ž2i. = ⭈⭈⭈ into ᑭ Žni., Ž i. which is the row stabilizer of Sq in ᑭ Žni.. We denote by ᑭ Žn,i.q the thus obtained Žparabolic. subgroup of ᑭ Žni.. Similarly, we denote by ᑭ Žn,i.y the Ž i. parabolic subgroup of ᑭ Žni. obtained as the column stabilizer of Sy . We set Ž2. Žr. ᑭ n , "s ᑭ Ž1. n , " = ᑭ n , " = ⭈⭈⭈ = ᑭ n , " .
Ž 5.1.1.
Then ᑭ n, " are parabolic subgroups of ᑭ n . We now define subgroups W"s W" Ž . of W by W"s ᑭ n, "h Ž⺪rr⺪. n. Then W" can be written as W"s W"Ž1. = W"Ž2. = ⭈⭈⭈ = W "Ž r . ,
Ž 5.1.2.
where W"Ž i. , ᑭ Žn,i. " h Ž⺪rr⺪. n i is a subgroup of W generated by ᑭ Žn,i. "
ARIKI ᎐ KOIKE ALGEBRAS
833
and t j with piy1 - j F pi . Next we consider the Hecke algebra version of these subgroups. Let Hn, " be the parabolic subalgebra of Hn which is specialized to ᑭ n, " by 0 . Then Hn, " can be written as Ž2. Žr. Hn , "s HnŽ1. , "m Hn , "m ⭈⭈⭈ m Hn , " ,
Ž 5.1.3.
where Hn,Ž i." is the subalgebra of Hn, " which is specialized to ᑭ Ž"i. . We now define an R1-algebra H " as the subalgebra of H h generated by Hn, " and 1 , . . . , n . Then it follows from 3.6 that H " can be written as H "s H "Ž1. m H "Ž2. m ⭈⭈⭈ m H "Ž r .,
Ž 5.1.4.
where H "Ž i. is the subalgebra of H h generated by Hn,Ž i." and j with piy1 - j F p. 5.2. In the construction of irreducible representations of HK given in Section 4, we consider the special case where s Ž Ž1., . . . , Ž r . . is such that Ž i. is a partition Ž n. of n, and other Ž j. are empty partitions. Then H Ž . s H h, and so Z s Z 0 gives the irreducible representation of HK . It follows that Z is a one dimensional representation of HK , whose character will be denoted by qŽ i.. Then we have
qŽ i. Ž a j . s q
Ž 2 F j F n. ,
qŽ i. Ž j . s u i
Ž 1 F j F n. .
Ž 5.2.1.
Next we consider the case where is such that Ž i. is a partition Ž1n . of n, and other Ž j. are empty partitions. Then again H Ž . s H h, and Z s Z 0 is a one dimensional representation, whose character will be denoted by qŽ i.. Then we have
qŽ i. Ž a j . s yqy1
Ž 2 F j F n. ,
qŽ i. Ž j . s u i
Ž 1 F j F n. .
Ž 5.2.2.
The characters qŽ i. and qŽ i. are realized over R. We denote by 1Ž i. Žresp. 1Ž i. . the linear character of W obtained from Ž i. Ž q resp. qŽ i. . by the specialization 0 : R ª ⺓. Note that 1Ž r . coincides with the identity character, and 1Ž1. coincides with the character det V , which associates to w g W the determinant det V Ž w . on V. Remark 5.3. The characters qŽ i. and qŽ i. are also expressed by using the standard generators a1 , . . . , a n4 . We have qŽ i. Ž a1 . s qŽ i. Ž a1 . s u i . In fact, by Ž3.3.3., the formula in Lemma 3.5 shows that S j coincides with Tj once jy1 and j are replaced by a scalar multiplication u i . This implies,
834
TOSHIAKI SHOJI
by Ž2.2.1., that T1 acts as 1 on the representation space of qŽ i. or qŽ i. inside of VKmn. The assertion follows from this. Ž i. 5.4. Let U be as in 3.1. For i s 1, . . . , r, we define elements ¨ q and in U by
Ž i. ¨y
Ž i. ¨q s
Ž i. ¨y s
Ý
q lŽ . T Ž ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. . ,
Ý
Ž yq . yl T Ž ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. . ,
gᑭ n
Ž .
gᑭ n
Ž 5.4.1.
Ž i. Ž Ž i. . where l Ž . denotes the length of g ᑭ n . Then ¨ q resp. ¨ y is a Ž i. Ž Ž i. . non-zero vector affording the linear character q resp. q of H h, Ž i. respectively. In fact, by the specialization 0 , ¨ q is mapped to an element Ž i. Ž i. mn Ž i. Ý g ᑭ n Ž ¨ n m ⭈⭈⭈ m ¨ 1 . in V which is clearly non-zero. Hence ¨ q is Ž i. also non-zero. Similarly ¨ y is non-zero. Now put
aqs
Ý
gᑭ n
q lŽ . a
ays
Ž . Ý Ž yq . yl a .
gᑭ n
Then it can be checked that a j aqs qaq and a j ays yqy1 ay for j s Ž i. Ž i. Ž i. Ž i. 2, . . . , n. Hence we have Tj ¨ q s q¨ q and Tj ¨ q s yqy1 ¨ y . On the other Ž i. mn Ž i. Ž i. hand, since ¨ q is contained in Vi , we have j ¨ q s u i ¨ q . Similarly, we Ž i. Ž i. Ž i. Ž Ž i. . have j ¨ y s u i¨ y . It follows that ¨ q resp. ¨ y generates an H h-module affording the character qŽ i. Žresp. qŽ i. .. By the specialization 0 , we have elements ¨ Ž"i. g U , which are given as Ž i. ¨q s
Ž i. ¨y s
Ý
Ž ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. . ,
Ý
Ž y1. l Ž ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. . .
gᑭ n
gᑭ n
Ž .
Ž 5.4.2.
Let i be the character of Ž⺪rr⺪. n which is the restriction of 1Ž i. or 1Ž i. on W Ži.e., the character such that i Ž t j . s i for j s 1, . . . , n.. Then ¨ nŽ i. m ⭈⭈⭈ m ¨ 1Ž i. is the image of ¨ 0 under the projection from V mn to the isotopic subspace for i . It follows that one can write ¨ Ž"i. s e Ž"i. ¨ 0 , where e Ž"i. are idempotents in ⺓W, up to non-zero scalar, affording the character 1Ž i. or 1Ž i., which are given as follows: Ž i. eq s ryn
Ý
1Ž i. Ž wy1 . w,
wgW Ž i. ey
yn
sr
Ý wgW
1Ž i. Ž wy1 . w.
Ž 5.4.3.
ARIKI ᎐ KOIKE ALGEBRAS
835
We now return to the case of H h. Since ¨ Ž"i. g UR 1, it follows from Ž i. Corollary 3.9 that there exist e Ž"i. g H h such that ¨ Ž"i. s e Ž"i. ¨ 0 . Then H heq h Ž i. h Ž i. Žresp. H ey . is an H -module affording the linear character q Žresp. Ž i. Ž i. Ž i. qŽ i. .. In particular, we have Ž e Ž"i. . 2 s ␥ " e " for some ␥ " g R1. We note Ž i. that ␥ " / 0. In fact, by the specialization 0 : R1 ª ⺓, e Ž"i. are mapped to the elements e Ž"i. in ⺓W, which are idempotents up to scalar from the Ž i. .y1 Ž i. Ž i. previous remark. Hence ␥ " is non-zero, and Ž␥ " e " give rise to h idempotents in H . Since e Ž"i. are idempotents up to scalar, e Ž"i.H h turn out to be right H h-modules affording the character qŽ i. or qŽ i.. In particular, for each w g W, we have Ž i. Ž i. Ž i. a w eq s eq a w s qŽ i. Ž a w . eq , Ž i. Ž i. Ž i. a w ey s ey a w s qŽ i. Ž a w . ey .
Ž 5.4.4.
5.5. Now the linear characters qŽ i., qŽ i. on Hn ih, r give rise to linear characters on H Ž i. under the isomorphism Hn ih, r , H Ž i.. We denote again by qŽ i. or qŽ i. their restriction to H "Ž i.. Let us define a linear character q Žresp. q . on Hq Žresp. Hy . by
q s qŽ1. m qŽ2. m ⭈⭈⭈ m qŽ r . , q s qŽ1. m qŽ2. m ⭈⭈⭈ m qŽ r . .
Ž 5.5.1.
Fixing i for the moment, we write Ž i. s s Ž 1 , 2 , . . . .. According to the decomposition WqŽ i. , W1 , r = W 2 , r = ⭈⭈⭈ , HqŽ i. is decomposed as Ž i. Hq , H1h, r m H 2h, r m ⭈⭈⭈ .
Ž 5.5.2.
Ž i. Then the linear character q Ž i. on Hq is decomposed as qŽ i. s q,Ž i.1 m q,Ž i.2 Ž i. Ž i. Ž i. m ⭈⭈⭈ , where q, j is nothing but q on Hhj, r as given in 5.2. Let eq, j be h Ž i. Ž . the idempotent up to scalar in H j, r corresponding to q, j as given in 5.4. Ž i. Ž i. Then under the isomorphism Ž5.5.2., eq, 1 m eq, 2 m ⭈⭈⭈ gives rise to an Ž i. idempotent Žup to scalar. in Hq corresponding to qŽ i., which will be Ž i. denoted by eq . In a similar way, according to the decomposition
HyŽ i. , HU1h, r m HU2h, r m ⭈⭈⭈ , where * s Ž U1 , U2 , . . . . is the dual partition of , qŽ i. is decomposed as Ž i. qŽ i. s q,Ž i.1 m q,Ž i.2 m ⭈⭈⭈ , and we define an idempotent Žup to scalar. ey Ž i. Ž i. Ž i. Ž i. corresponding to q as ey s ey, 1 m ey, 2 m ⭈⭈⭈ . We now define elements e " in H "s H "Ž1. m ⭈⭈⭈ m H "Ž r . by Ž2. Žr. e "s e Ž1. " m e " m ⭈⭈⭈ m e " .
Ž 5.5.3.
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TOSHIAKI SHOJI
Then Ž e " . 2 s ␥ " e " with non-zero elements ␥ "g R1. Hence ␥y1 " e " are idempotents in H " affording the linear character q or q . Moreover, if we define e "g ⺓W " by Ž2. Žr. e "s e Ž1. " m e " m ⭈⭈⭈ m e " ,
where e Ž"i. are idempotents in ⺓W Ž i. affording linear characters 1Ž i. or 1Ž i., then e " are idempotents Žup to scalar. affording the linear characters 1 or 1 of W". Under the specialization 0 , e " are mapped to e "g ⺓W". ŽThe linear characters 1 , 1 are defined in a similar way as in the case H ".. We note that H heq, H h mHq q , and H hey, H h mHy q . In fact this follows, by the specialization 0 , from the corresponding facts for ⺓W. ŽSee the proof of Theorem 4.3.. We have the following lemma. LEMMA 5.6.
Let e " be as in 5.5. Then dim K Hom HK Ž HK e " , HK e . . s 1.
Proof. For partitions s Ž 1 , 2 , . . . . and s Ž 1 , 2 , . . . . of n, we define a partial order G by the condition that Ýiks 1 k G Ýiks1 k for i s 1, 2, . . . . Then for s Ž Ž1., . . . , Ž r . ., s Ž Ž1., . . . , Ž r . . g Pn, r , we define a partial order G by the condition that < Ž i. < s < Ž i. < and that Ž i. G Ž i. for each 1 F i F r. In order to prove the lemma, it is enough to show the following statement. Ž5.6.1. If an irreducible representation Z occurs in Hk eq Žresp. HK ey ., then G Žresp. F .. Moreover, the multiplicity of Z in HK e " is equal to one. We show Ž5.6.1.. First consider the q case. Let ˜qŽ i. be the linear character of Hn,Ž i.q obtained by restricting qŽ i. from HqŽ i. to Hn,Ž i.q. Then HKŽ i. mHq,Ž i.K qŽ i. is isomorphic to the Hn,Ž i.K-module Hn,Ž i.K mH n,Ž i.q, K ˜qŽ i., extended to the HKŽ i.-module by defining the action of j Ž i. g H by the scalar multiplication u i . In fact, this is easily checked by realizing the corresponding representation of Hn i , K inside of V mn i as in the proof of Theorem 4.3. Let be a partition of n i . Now it is well known that an irreducible Hn,Ž i.K-module Z occurs in Hn,Ž i.K mH n,Ž i.q, K ˜qŽ i. only when G Ž i., and ZŽ i. occurs exactly once. The assertion Ž5.6.1. follows from this by virtue of Theorem 4.3. The y case is dealt with similarly, by using the fact that Z occurs in Hn,Ž i.K mH n,Ž i.y, K ˜qŽ i. only when F Ž i. and ZŽ i. occurs exactly once. ŽHere ˜qŽ i. is the character of Hn,Ž i.y obtained as the restriction of qŽ i. to Hn,Ž i.y.. Thus Ž5.6.1. is proved, and the lemma follows.
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5.7. Let us take non-zero intertwining operators f "g Hom H K Ž HK e . , HK e " . . Then by Lemma 5.6, f " is unique up to scalar, and the image of f " is isomorphic to the irreducible HK-module Z . We determine the operator fq explicitly. Since Ž␥ " .y1 e " are idempotents affording the characters q or q of H ", K , one can write y1 y1 fq Ž ey . s ␥q ␥y ey fq Ž ey . eq .
Let X be a complete set of representatives in W for the double cosets Wy_ WrWq. Then we have Ž5.7.1. The set ey a w eqg HK ¬ w g X, ey a w eq/ 04 gives a basis of the space ey HK eq. In fact, by using the defining relations in 3.6, one can check that a w 1 w w 2 g Hy a w Hq for w 1 g Wy, w 2 g Wq. Hence by Ž5.4.4., ey HK eq is generated by ey a w eq¬ w g X 4 . Note that by the specialization 0 , ey H heq is mapped to ey ⺓Weq, whose basis is given by ey weq¬ w g X, ey weq/ 04 . In particular, ey a w eq/ 04 are linearly independent over R1. Ž5.7.1. follows from this. We have Wy_ WrWq, ᑭ n, y_ ᑭ nrᑭ n, q, and ᑭ n, " are parabolic subgroups in ᑭ n . Hence we can choose X as the subset of ᑭ n consisting of distinguished representatives for the double cosets ᑭ n, y_ ᑭ nrᑭ n, q; i.e., we fix X as the set of g ᑭ n such that is a minimal length element in the double coset ᑭ n, y ᑭ n, q. Thus fq Ž ey . g ey HK eq can be written as fq Ž ey . s Ý c ey a eq gX
with c g K. For standard tableaux ⺣1 and ⺣ 2 of shape , we denote by Ž⺣1 , ⺣ 2 . the element in ᑭ n which maps each entry in ⺣ 2 to the corresponding entry in ⺣ 1. Put 0 s Ž⺣y, ⺣q .. We note that 0 g X. In fact, since 0 g ᑭ nŽ ., we are reduced to the case where r s 1, and the claim is easily checked in this case Žcf. wG, 2x.. We have the following lemma. LEMMA 5.8. Assume that g X. Then ey a eqs 0 except the case where s 0 . In particular, fq Ž ey . coincides with ey a 0 eq up to a non-zero scalar in K. Proof. First we note that the following formula holds in HK . The proof is straightforward from Ž3.6.6. and Ž3.6.7..
k a eqs a k ⬘ eqq
Ý
⬘-
c ⬘ a ⬘ eq ,
for 1 F k F n, g ᑭ n , where k⬘ s y1 Ž k . and c ⬘ g K.
Ž 5.8.1.
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TOSHIAKI SHOJI
Next we show that ey a eqs 0 unless g ᑭ nŽ . l X. Assume that f ᑭ nŽ .. Then there exists t j g W Ž c1 . and t k g W Ž c 2 . with c1 / c2 such that t j y1 s t k . In view of Ž5.8.1., ey k a eq can be written as ey k a eqs ey a j eqq
Ý
⬘-
c ⬘ ey a ⬘ eq .
In this equation, we may assume that ⬘ g X. It follows, by Ž5.4.4., that we have
Ž u c 2 y u c1 . ey a eqs
Ý
⬘-
c ⬘ ey a ⬘ eq .
But this implies ey a eqs 0, in view of Ž5.7.1.. We now assume that g ᑭ nŽ . l X. Then the assertion of the is easily reduced to the case where r s 1, i.e., the case of the algebra Hn of type A ny1. But this is exactly the case discussed by i.e., he proved in wG, pp. 845᎐846x the corresponding fact for Hn . our assertion follows from his result. Thus the lemma is proved.
lemma Hecke Gyoja; Hence
5.9. Let ⺣ be a standard tableau of shape . For each ⺣, we shall construct an idempotent in HK affording the irreducible representation Z , following the strategy employed in wGx. We set " Ž⺣. s Ž⺣, ⺣ " ., and denote by a " Ž⺣. the elements a for s " Ž⺣.. Since y Ž⺣.y1q Ž⺣. s 0 , the element ay Ž⺣.y1 aq Ž⺣. coincides with a 0 modulo Ž q y qy1 . HK . Hence by Lemma 5.8, we see that ey ay Ž⺣.y1 aq Ž⺣. eq coincides with ey a 0 eq up to a non-zero scalar. By taking fq appropriately according to ⺣, we may assume that fq Ž ey . s ey ay Ž ⺣ .
y1
aq Ž ⺣ . eq .
Ž 5.9.1.
A similar argument also works for fy, and we have fy Ž eq . s eq aq Ž ⺣ .
y1
ay Ž ⺣ . ey .
Ž 5.9.2.
Now by Schur’s lemma, applied for fq Ž HK ey . , Z , one can write fq fy fq Ž ey . s cfq Ž ey . with some c s cŽ⺣. g K. Then by Ž5.9.1. and Ž5.9.2., we have y1 y1 y1 y1 ey ay aq eq aq ay ey ay aq eqs cey ay aq eq , y1 where a "s a " Ž⺣.. If we set eŽ⺣. s ay ey ay1 y aq eq aq , the above formula implies that 2
e Ž ⺣. s c ⭈ e Ž ⺣. .
Ž 5.9.3.
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839
We claim that Ž5.9.4. c is a non-zero element in K which is mapped to c g ⺓ y 04 via the specialization 0 . Let e "g ⺓W be as in 5.5. By the specialization 0 , the element eŽ⺣. is y1 mapped to an element eŽ⺣. s y ey y1 y q eq q g ⺓W, where "s 2 Ž⺣, ⺣ " .. We have the equality eŽ⺣. s ceŽ⺣. with c g ⺓. Hence in order to show the claim, it is enough to see that eŽ⺣. is an idempotent up to a non-zero scalar. Here we show a more general fact. The following lemma implies the claim Ž5.9.4.. LEMMA 5.10. The element eŽ⺣. is an idempotent in ⺓W up to a non-zero scalar. Moreo¨ er, if ⺣ and ⺣⬘ are standard tableaux of shape such that ⺣ / ⺣⬘, then we ha¨ e eŽ⺣. eŽ⺣⬘. s 0. Proof. The first assertion is easily reduced to the case where r s 1, i.e., to the case where W"s W"Ž i. and ⺣ s S Ž i. is a standard tableau of shape Ž is a partition of n.. Hence e "s e Ž"i. are idempotents Žup to scalar. y1 affording the linear character 1Ž i. or 1Ž i. of W". Then q Wq q , ᑭ n, n y1 n h Ž⺪rr⺪. and y Wy y , ᑭ n, * h Ž⺪rr⺪. , where ᑭ n, Žresp. ᑭ n, * . is the row stabilizer Žresp. the column stabilizer. of S Ž i. in ᑭ n . Let SŽ i. be y1 the linear character of q Wq q obtained from 1Ž i. by the conjugation y1 by q, and similarly the linear character SŽ i. of y Wy y is defined. Ž i. y1 y1 Then " e " " are the idempotents in " ⺓W" " affording the linear character SŽ i. or SŽ i. Žup to scalar.. Then, as in Ž5.4.3., we have yn q eq y1 q sr
Ý
SŽ i. Ž w . w.
wg qW q y1 q
Let i be the linear character of Ž⺪rr⺪. n obtained from 1Ž i. or 1Ž i. as before. If we put eq Ž S . s
Ý
gᑭ n,
,
e0 s ryn
Ý
i Ž w . w,
wg Ž⺪rr ⺪ . n
Ž . Ž . we have q eq y1 q s e 0 eq S s eq S e 0 . A similar argument shows that Ž . Ž . y ey y1 s e e S s e S e , where ey Ž S . s Ý g ᑭ n, *Žy1. lŽ . . This y 0 y y 0 implies that e Ž S . s e0 ey Ž S . ⭈ e0 eq Ž S . s e0 ey Ž S . eq Ž S . ,
Ž 5.10.1.
since e02 s e0 and e0 commutes with e " Ž S .. Here we note that ey Ž S . eq Ž S . is the Young symmetrizer in ᑭ n associated to the standard tableau S, and so is the idempotent up to scalar. Thus the first assertion of Lemma 5.10 follows.
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TOSHIAKI SHOJI
For the second assertion, it is enough to show that eq q Ž ⺣ .
y1
y Ž ⺣⬘ . eys 0.
Ž 5.10.2.
But a similar statement as in Lemma 5.8 holds by replacing eq with ey, and 0 with Ž⺣q, ⺣y .. Hence, by the specialization 0 , we see that eq eys 0 unless g ᑭ n, q Ž⺣q, ⺣y . ᑭ n, y. Since q Ž⺣.y1y Ž⺣⬘. f ᑭ n, q Ž⺣q, ⺣y . ᑭ n, y, we obtain Ž5.10.2.. 5.11. By Ž5.9.4. and by Lemma 5.10, cŽ⺣.y1 eŽ⺣. is an idempotent in HK , which is specialized by 0 to an idempotent cŽ⺣.y1 eŽ⺣. in ⺓W. We note that eŽ⺣. is an idempotent affording the irreducible representation Z since HK a " e " ay1 " , HK e " as HK -modules. In particular, we have eŽ⺣. eŽ⺣⬘. s 0 if ⺣ and ⺣⬘ have different shape. By the specialization 0 , a similar formula holds also for eŽ⺣.. Hence, by combining with Lemma 5.10, we have Ž5.11.1. The set cŽ⺣.4 gives a complete set of mutually orthogonal primitive idempotents in ⺓W, where ⺣ runs over all the standard tableaux of any shape in Pn, r . Returning to the setting for HK , we consider the idempotents eŽ⺣. and Ž e ⺣⬘. in the case where ⺣ and ⺣⬘ are standard tableaux of the same shape. The following lemma is a generalization of Lemma 2.3.1 in wGx. LEMMA 5.12. Let ⺣, ⺣⬘ be standard tableaux of shape such that ⺣ / ⺣⬘. Put y Ž⺣. s 1 1X Ž resp. y Ž⺣⬘. s 2 2X ., where 1 Ž resp. 2 . is the minimal length element in the coset y Ž⺣. ᑭ nŽ . Ž resp. y Ž⺣⬘. ᑭ nŽ .., and 1X , 2X g ᑭ nŽ .. Assume that l Ž 1 . F l Ž 2 . if 1 / 2 , and assume that l Ž 1X . G l Ž 2X . if 1 s 2 . Then we ha¨ e eŽ⺣. eŽ⺣⬘. s 0. Proof. As discussed in 5.10, it is enough to show that eq aq Ž ⺣ .
y1
ay Ž ⺣⬘ . eys 0.
Ž 5.12.1.
Let ⺣ s Ž S Ž1., . . . , S Ž r . .. Then it is easy to see that 1 is written as
1 s
ž
⭈⭈⭈
piy1 q 1
piy1 q 2
⭈⭈⭈
pi
⭈⭈⭈
⭈⭈⭈
k 1Ž i.
k 2Ž i.
⭈⭈⭈
k nŽ i.i
⭈⭈⭈
/
, Ž 5.12.2.
where k 1Ž i. - k 2Ž i. - ⭈⭈⭈ - k nŽ i.i are the letters occurring in the entries of the tableau S Ž i.. This implies that 1 is also the minimal length element in the coset q Ž⺣. ᑭ nŽ ., and so one can write q Ž⺣. s 1 1Y with 1Y g ᑭ nŽ .. Hence we have aq Ž⺣. s a 1 a 1Y , and ay Ž⺣⬘. s a 2 a 2X . We note that, by virtue of Ž5.12.2., there exist standard tableaux ⺣ 0 , ⺣X0 such that 1Y s q Ž⺣ 0 ., 1X s y Ž⺣ 0 . and that 2X s y Ž⺣X0 ..
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First we assume that l Ž 1 . F l Ž 2 . with 1 / 2 . Let Y s 2 g ᑭ n ¬ 4 . Then it is easy to see that ay1 a can be written as F y1 1 1 2 ay1 a 2 s 1
Ý cy ay
Ž cy g K . .
Ž 5.12.3.
ygY
We note that y f ᑭ nŽ .. In fact, if y s 2 g ᑭ nŽ ., then 2 g y1 ᑭ nŽ .. It follows that l Ž y1 . G l Ž 2 ., and by our assumption, l Ž y1 . y1 Ž . G l Ž 1 .. Combined with F y1 1 , we have s 1 . But then 1 ᑭ n s 2 ᑭ nŽ . and this contradicts the assumption that 1 / 2 . Now it follows from Ž5.12.3. that we can write eq aq Ž ⺣ .
y1
ay Ž ⺣⬘ . eys
Ž c g K . ,
Ý c eq a ey
where g ᑭ n y ᑭ nŽ .. A similar argument as in the proof of Lemma y1 5.8 shows that eq a eys 0 unless g ᑭ n, q y1 0 ᑭ n, y, where 0 s y1 Ž⺣q, ⺣y .. Since 0 g ᑭ nŽ ., we get Ž5.12.1.. Next assume that l Ž 1X . G l Ž 2X . with 1 s 2 . Then we have aq Ž ⺣ .
y1
Y a X s a ay Ž ⺣⬘ . s ay1 2 q Ž ⺣0 . 1
y1
ay Ž ⺣X0 . ,
with ⺣ 0 / ⺣X0 . This is essentially the same situation as discussed in wG, Lemma 2.3.1x, and the argument there works well for our case. For the sake of completeness, we give the proof below. For any standard tableau ⺣ such that q Ž⺣. g ᑭ nŽ ., we have l Ž q Ž ⺣ . . q l Ž y Ž ⺣ . . s l Ž y1 0 .. By applying this formula with ⺣ s ⺣ 0 , and by our assumption, we have X l Ž y1 0 . G l Ž q Ž ⺣ 0 . . q l Ž y Ž ⺣ 0 . . .
Ž 5.12.4.
Now aq Ž⺣ 0 .y1 ay Ž⺣X0 . can be written as the form Ý y g Y ⬘ c y a y with c y g K, where Y ⬘ s x y Ž⺣X0 . ¬ x F q Ž⺣ 0 .y1 4 . Since eq a y eys 0 unless y g Ž .y1y Ž⺣X0 . by Ž5.12.4.. ᑭ n, q y1 0 ᑭ n, y, we have eq a y eys 0 if y / q ⺣ 0 X . s l Ž y .. Then y f Assume that y s q Ž⺣ 0 .y1y Ž⺣ 0 . and that l Ž y1 0 y1 y1 ᑭ n, q 0 ᑭ n, y since 0 is the unique minimal length element in this Ž . double coset, and y / y1 0 . Hence 5.12.1 holds also in this case. The lemma is now proved. 5.13. Let T be the set of standard tableaux of shape . We give a total order ⺣ G ⺣⬘ in T , compatible with the property given in Lemma 5.12, i.e., the order satisfying the following properties: assume that ⺣ ) ⺣⬘, and put y Ž⺣. s 1 1X and y Ž⺣⬘. s 2 2X as in Lemma 5.12. Then we
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TOSHIAKI SHOJI
require that l Ž 1 . F l Ž 2 . if 1 / 2 , and l Ž 1X . G l Ž 2X . if 1 s 2 . Clearly such an order exists. We give a numbering on T as T s ⺣ 1 , ⺣ 2 , . . . , ⺣ f 4 along this order. Then by Lemma 5.12, we see that eŽ⺣ i . eŽ⺣ j . s 0 if i ) j. We define an element ˜ eŽ⺣ i . g HK by y1 y1 y1 ˜e Ž ⺣ i . s Ž 1 y cy1 1 e Ž ⺣ 1 . .Ž 1 y c 2 e Ž ⺣ 2 . . ⭈⭈⭈ Ž 1 y c iy1 e Ž ⺣ iy1 . . c i e Ž ⺣ i . ,
with c i s cŽ⺣ i .. It is proved in Lemma 4.4 in wGx that if eŽ⺣1 ., . . . , eŽ⺣ f .4 is a set of idempotents satisfying the above properties, then ˜ eŽ⺣. < ⺣ g T 4 gives a family of mutually orthogonal idempotents. ŽIt is also shown in wGx that ˜ eŽ⺣. does not depend on the choice of the total order in T .. It is clear that ˜ eŽ⺣. is a primitive idempotent affording the irreducible module Z . It follows from Lemma 5.10 that, by the specialization 0 , ˜ eŽ⺣. is mapped to the idempotent cŽ⺣.y1 eŽ⺣. g ⺓W. Summing up the above arguments, we have the following theorem. THEOREM 5.14. The set ˜ eŽ⺣. g HK ¬ ⺣ g T , g Pn, r 4 gi¨ es a complete family of mutually orthogonal primiti¨ e idempotents in HK . By the specialization 0 , ˜ eŽ⺣. is mapped to the primiti¨ e idempotent ˜ cŽ⺣.y1 eŽ⺣. g ⺓W, which forms a complete family of mutually orthogonal primiti¨ e idempotents in ⺓W.
6. A FROBENIUS FORMULA FOR THE CHARACTERS OF HK 6.1. In this section, we introduce indeterminates x Žj i. Ž1 F i F r, 1 F j F m i . associated to the basis ¨ jŽ i. of V. We also write these indeterminates as x 1 , . . . , x m , corresponding to the basis ¨ 1 , . . . , ¨ m . Let s Ž Ž1., . . . , Ž r . . be a multipartition in Pn, r . Following wM. I. Appendix Bx, we extend the notion of power sum symmetric functions and Schur functions to the case of multipartitions. Recall that is a primitive r th root of unity. We define a ‘‘power sum function’’ P associated to as follows. For each integer k G 1 and i such that 1 F i F r, put r
PkŽ i. Ž x . s
Ý yi j pk Ž x Ž j. . , js1
where pk Ž x Ž j. . denotes the kth power sum symmetric function with respect Ž j. to the variables x 1Ž j., . . . , x m . For a partition Ž i. s Ž Ž1i., Ž2i., . . . , Žki.i . of n i , j Ž . Ž i. we define a function P x by ki
PŽ i. Ž x . s
Ł PŽ i. Ž x . , js1
Ž i. j
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and define a function PŽ x . by r
P Ž x . s
Ł P Ž x . . Ž i.
Ž 6.1.1.
is1
We denote by PŽ x . the complex conjugate of PŽ x .. Next we define a ‘‘Schur function’’ S Ž x . associated to by r
S Ž x . s
Ł s Ž x Ž i. . , Ž i.
Ž 6.1.2.
is1
where sŽ i. Ž x Ž i. . denotes the Schur function associated to the partition Ž i. Ž i. with respect to the variables x 1Ž i., . . . , x m . i The set of conjugacy classes in W is parametrized by Pn, r . The correspondence is given as follows. Let w s t 1c1 ⭈⭈⭈ t nc n g W. Now g ᑭ n is written as a product of disjoint cycles. Let z s Ž i1 , i 2 , . . . , i k . be a cycle occurring in the decomposition of . We say that z is of type a if c i1 q c i 2 q ⭈⭈⭈ qc i k ' a Žmod r ., Ž1 F a F r .. Then w belongs to the conjugacy class corresponding to s Ž Ž1., . . . , Ž r . ., if Ž i. is defined as the product of cycles of type i. For each g Pn, r , we fix a representative w of the conjugacy class in W corresponding to as follows. First we put w Ž n, i . s t ni sn sny1 ⭈⭈⭈ s2 . Then w Ž n, i . corresponds to an n-cycle of type i. For a partition s Ž 1 , 2 , . . . . of n, we define w Ž , i . g W by w Ž , i . s w Ž 1 , i . = w Ž 2 , i . = ⭈⭈⭈ under the natural embedding W1 , r = W 2 , r = ⭈⭈⭈ into W. Let s Ž Ž1., . . . , Ž r . . g Pn, r . Under the isomorphism W Ž i. , Wn , r , we may regard i w Ž Ž i., i . as an element in W Ž i.. Then we define w g W Ž . ; W as w s w Ž Ž1. , 1 . w Ž Ž2. , 2 . ⭈⭈⭈ w Ž Ž r . , r . .
Ž 6.1.3.
The above construction of w is generalized as follows. Let ⌺ be the subset of W consisting of w s t 1c1 ⭈⭈⭈ t nc n such that any k-cycle 1 occurring in the decomposition of is of the form 1 s s pqky2 ⭈⭈⭈ s pq1 s p a for some p G 2, and w is a product of the factors t pq ky2 s pqky2 ⭈⭈⭈ s pq1 s p . We write as ⌺ the set of elements of ⌺ which is conjugate to w . Then w g ⌺ is obtained in a similar way as w by using a suitable embedding WŽ1. = WŽ1. = ⭈⭈⭈ into W. 1 ,r 2 ,r Let 1 be the irreducible character of W corresponding to g Pn, r . Then it is known by wM, Appendix B, Ž9.5.x that the formula S s
Ý
g Pn, r
zy1 1 Ž w . P
holds, where z is the order of the centralizer of w in W.
Ž 6.1.4.
844
TOSHIAKI SHOJI
6.2. Let K ⬘ s K Ž x 1 , . . . , x m . be the field of rational functions on K. In the remainder of this paper, we assume that Uq Ž ᒄ ., H , V, etc., are defined over K ⬘. ŽBy abbreviation, we use the notations such as H , V, etc., instead of HK ⬘ , VK ⬘ , etc.. Let Cn, m be the set of m-tuples of integers c i G 0 such that Ýc i s n. Take c s Ž c1 , . . . , c m . g Cn, m . For a Uq Ž ᒄ .-module M, we denote by Mc the weight subspace of M with weight c; i.e., q j acts on Mc by the scalar multiplication by q c j. We consider the H m Uq Ž ᒄ .-module V mn. For each sequence c as above, let Ec be the projection from V mn to the weight subspace Vcmn of V mn. We define an operator D on V mn by Ds
Ý x c Ec , c
where c runs over all the elements in Cn, m , and x c denotes x 1c1 x 2c 2 ⭈⭈⭈ x mc m . Note that weight spaces in V mn are left invariant by the action of H . Hence Ec commutes with the action of H , and so D also commutes with H . We also note that Vcmn coincides with the subspace of V mn generated by ¨ i1 m ¨ i 2 m ⭈⭈⭈ m ¨ i n such that 噛 i k < bŽ i k . s j4 s c j . Let ⺓⬘ s ⺓Ž x 1 , . . . , x m .. In a similar way as above, we regard V as defined over ⺓⬘, and consider the ⺓⬘W m UŽ ᒄ .-module V mn. We define an operator ⌺ c x c Ec on V mn, which we denote also by D, where Ec is the projection from V mn to the weight subspace of weight c. The trace on V mn of elements in W multiplied by D is given as follows. The proof is a straightforward computation. LEMMA 6.3. Ži.
Let w Ž n, i . be as in 6.1. Then we ha¨ e
Tr Ž Dw Ž n, i . , V mn . s
m
Ý bŽ j.i x jn s PnŽ i. Ž x . . js1
Žii.
More generally, for w g ⌺ , we ha¨ e Tr Ž Dw, V mn . s P Ž x . .
6.4. Let p be a primitive idempotent in ⺓W affording the irreducible representation Z . Then we have Tr Ž Dp , V mn . s S Ž x . .
Ž 6.4.1.
In fact, the proof of Lemma 3.7 in Ram wRx can be applied to our case also. By using the argument there, we have Tr Ž Dp , V mn . s < W
Ý wgW
1 Ž w . Tr Ž Dw, V mn . .
ARIKI ᎐ KOIKE ALGEBRAS
845
Since D commutes with the action of W, TrŽ Dw . only depends on the conjugacy class containing w. Hence, by using Lemma 6.3, Žii., we have Tr Ž Dp , V mn . s < W
Ý
g Pn, r
< C < 1 Ž w . P Ž x . ,
where < C < is the number of elements in the conjugacy class C of W corresponding to . The last formula coincides with S Ž x . by Ž6.1.4.. Thus Ž6.4.1. holds. 6.5. We now pass to the setting for H and consider the H m Uq Ž ᒄ .module V mn. Let q be the irreducible character of H corresponding to . We have the following proposition. PROPOSITION 6.6.
For any h g H , we ha¨ e
Tr Ž Dh, V mn . s
Ý
g Pn, r
q Ž h . S Ž x . .
Proof. The proof is done in a similar way as in the proof of Theorem 3.8 in wRx, once we admit Theorem 5.14. By Theorem 5.14, there exists a complete family of mutually orthogonal primitive idempotents pi ¬ g Pn, r , 1 F i F d 4 of H Žhere pi affords the irreducible module Z with d s dim Z . such that each pi is mapped, by the specialization 0 , to a primitive idempotent pi in ⺓W affording the irreducible module Z . By a standard argument Žcf. wR, Ž3.8.x., we can write Tr Ž Dh, V mn . s
Ý h ii Tr Ž Dpi , V mn . ,
Ž 6.6.1.
, i
where h ii is remarked in and so is left
the diagonal element of the matrix of h on Z . Now as wR, Ž3.5.x, TrŽ Dpi . is independent of indeterminates q, u i , invariant by the specialization 0 . Therefore we have Tr Ž Dpi , V mn . s Tr Ž Dpi , V mn . s S Ž x . .
Ž 6.6.2.
The second identity follows from Ž6.4.1.. By substituting Ž6.6.2. into Ž6.6.1., we get the required formula. ŽThere exists a simple, direct proof of Proposition 6.6 without using Theorem 5.14, suggested by the referee, as follows. Since D g ŽUq Ž ᒄ .., by virtue of Theorem 2.4, it is enough to show that TrŽ D, V . s S Ž x .. But Uq Ž ᒄ . , Uq Ž gl m 1 . m ⭈⭈⭈ m Uq Ž gl m r . and according to this isomorphism, V is decomposed as V , VŽ1. m ⭈⭈⭈ m VŽ r . , where VŽ i. is an irreducible Uq Ž gl m i .-module corresponding to the partition Ž i.. Then TrŽ D, V . can be written as r
Tr Ž D, V . s
Ł Tr Ž D Ž i. , V . , Ž i.
is1
846
TOSHIAKI SHOJI
where D Ž i. is the operator on Vimn i defined analogous to D. Using the corresponding result for the case Uq Ž gl m i . wR, Theorem 3.8x, we see that TrŽ D Ž i., VŽ i. . s sŽ i. Ž x Ž i. .. This implies that TrŽ D, V . s S Ž x . by Ž6.1.2., and the proposition follows.. 6.7. For g Pn, r , we define an element a g H imitating the definition of w in 6.1 as follows. First we put aŽ n, i . s ni a n a ny1 ⭈⭈⭈ a2 . Then, for each partition s Ž 1 , 2 , . . . . of n, we define a Ž , i . s a Ž 1 , i . m a Ž 2 , i . m ⭈⭈⭈ under the embedding H1 , r m H 2 , r m ⭈⭈⭈ into H . Let s Ž Ž1., . . . , Ž r . . g Pn, r . We define a g H Ž . ; H by a s a Ž Ž1. , 1 . a Ž Ž2. , 2 . ⭈⭈⭈ a Ž Ž r . , r . , where as in the group case, aŽ Ž i., i . is regarded as an element in H Ž i. via the isomorphism H Ž i. , Hn i , r . More generally, the element a w for w g ⌺ is also constructed in a similar way as above by using a suitable embedding of HŽ1. m HŽ1. m ⭈⭈⭈ into H . 1 ,r 2 ,r In general, for each w s t 1c1 t 2c 2 ⭈⭈⭈ t nc n g W, we set a w s 1c1 2c 2 ⭈⭈⭈ nc n a . Then, a w ¬ w g W 4 gives rise to a basis of H . We note that the above construction of a w for w g ⌺ is compatible with this definition. By the specialization 0 , a w is mapped to w. In particular, a is mapped to w. We shall compute the trace of Da w for w g ⌺ on V mn. The special case where corresponds to an n-cycle of type i is given explicitly. In the following, we give a slightly general formula. THEOREM 6.8.
Let a w s 1c1 2c 2 ⭈⭈⭈ nc n a n a ny1 ⭈⭈⭈ a2 . Then we ha¨ e
Tr Ž Da w , V mn . s
Ý u bŽc i . u bŽc i . 1
2
1
2
cn eŽ I . ⭈⭈⭈ u bŽ Ž q y qy1 . in. q
lŽ I .
x i1 x i 2 ⭈⭈⭈ x i n ,
I
Ž 6.8.1. where the sum is taken o¨ er all the sequences I s Ž i1 , i 2 , . . . , i n . such that 1 F i1 F i 2 F ⭈⭈⭈ F i n , eŽ I . Ž resp. l Ž I .. denotes the number of j such that i j s i jq1 Ž resp. i j - i jq1 .. Proof. The computation of the action of Da n a ny1 ⭈⭈⭈ a2 is completely the same as the proof of Theorem 4.1 in wRx. We have only to note that in this computation, the contribution of the term x i1 x i 2 ⭈⭈⭈ x i n comes from the basis vector ¨ i1 m ¨ i 2 m ⭈⭈⭈ m ¨ i n of V mn, on which k acts as a scalar multiplication by u bŽ i k . . Note that our generator a i g Hn is different from the one in wRx by scalar, and so the expression involving q in our formula is different from wRx.
ARIKI ᎐ KOIKE ALGEBRAS
847
6.9. More generally, we shall compute the trace of Da w for w g ⌺ . The cycle of type i corresponding to Žj i. is expressed as t ki sk sky1 ⭈⭈⭈ sl for some integers l F k such that Žj i. s k y l q 2. Put aŽ Žj i. . s ki a k a ky1 ⭈⭈⭈ a l . Then a can be written as the product of aŽ Žj i. . for various i, j. Note that the operator aŽ Žj i. . affects only the l y 1, l, . . . , k factors of V mn, which are disjoint for distinct Žj i.. Moreover, D is also decomposed according to such a decomposition of factors in V mn. Hence TrŽ Da . can be computed as the product of the traces TrŽ DaŽ Žj i. ... But the latter one coincides with the trace of DaŽ Žj i., i . on the Žj i.-fold tensor space of V. Therefore we have the following formula. PROPOSITION 6.10.
Let w g ⌺ . Then we ha¨ e
Tr Ž Da w , V mn . s
r
l Ž Ži. .
Ł Ł
is1 js1
Tr Da Ž Žj i. , i . , V m j . Ž i.
ž
/
In particular, the trace TrŽ Da w . coincides with the trace TrŽ Da . for any w g ⌺. 6.11. We now introduce a ‘‘monomial function’’ m Ž x . associated to g Pn, r by r
mŽ x . s
Ł m Ž x Ž i. . ,
Ž 6.11.1.
Ž i.
is1
where mŽ i. Ž x Ž i. . is a monomial symmetric function corresponding to the Ž i. partition Ž i. with respect to the variables x 1Ž i., . . . , x m . i Ž i. Ž mn Let qn x; q, u. s TrŽ DaŽ n, i ., V .. Then, thanks to Theorem 6.8, qnŽ i. Ž x; q, u. can be written as qnŽ i. Ž x ; q, u . s
Ý
g Pn, r
ui q nylŽ. Ž q y qy1 .
l Ž .y1
m Ž x . , Ž 6.11.2.
where l Ž . s Ý ris1 l Ž Ž i. . for s Ž Ž1., . . . , Ž r . ., and u denotes u k for the largest number k such that Ž k . is not an empty partition. Let qr Ž x 1 , . . . , x m ; t . be the Hall᎐Littlewood function Žcf. wM, III, 2x., which is defined by q0 Ž x ; t . s 1, m
qr Ž x ; t . s Ž 1 y t .
Ý x ir Ł is1
j/i
x i y tx j xi y x j
Ž r G 1. .
The following formula is obtained from the formula Žb. in wR, Theorem 4.13x:
Ý
&n
q nylŽ . Ž q y qy1 .
l Ž .y1
m Ž x . s
qn q y qy1
qn Ž x ; qy2 . . Ž 6.11.3.
848
TOSHIAKI SHOJI
Let c s Ž n1 , n 2 , . . . , n r . g Cn, r . If s Ž Ž1., . . . , Ž r . . g Pn, r is such that < Ž i. < s n i , we write g c. The following formula expresses the function qnŽ i. Ž x; q, u. in terms of Hall᎐Littlewood functions. PROPOSITION 6.12. qnŽ i. Ž x ; q, u . s
qn
r
q y qy1
u ci Ł qn jŽ x Ž j. ; qy2 . ,
Ý
js1
cg Cn, r
where c g Cn, r is written as c s Ž n1 , n 2 , . . . , n r ., and u c denotes u k for the largest integer k such that n k / 0. Proof. Since u is independent for g c for a fixed c g Cn, r , which is equal to u c , we can write qnŽ i. Ž x ; q, u . s
Ý
u ci
cg Cn, r
Ý
q nylŽ. Ž q y qy1 .
gc
s Ž q y qy1 .
r
ry1
u ci Ł
Ý
= Ž q y qy1 .
l Ž Ž j. .y1
qn q y qy1
Ý
m Ž x .
q n iylŽ
Ž j.
.
js1 Ž j.&n i
cg Cn, r
s
l Ž .y1
mŽ j. Ž x Ž j. .
r
u ic Ł qn jŽ x Ž j. ; qy2 . .
Ý
js1
cg Cn, r
The last equality follows from Ž6.11.3.. Thus the proposition is proved. 6.13. Let qnŽ i. Ž x; q, u. be as in Ž6.11.2.. For a multipartition s Ž Ž1., . . . , Ž r . . g Pn, r , we define a function q˜Ž x 1 , . . . , x m ; q, u. by l Ž Ži. .
r
q˜ Ž x 1 , . . . , x m ; q, u . s
Ł Ł
is1
js1
qŽ i.Žj i. Ž x ; q, u . .
Then by combining Proposition 6.6, Theorem 6.8, and Proposition 6.10, we obtain the following Frobenius type formula for the characters of H , which is a generalization of Theorem 4.14 in wRx. THEOREM 6.14.
For each g Pn, r , we ha¨ e q˜ Ž x ; q, u . s
Ý
g Pn, r
q Ž a . S Ž x . .
Ž 6.14.1.
ARIKI ᎐ KOIKE ALGEBRAS
849
As a corollary, we have the following. COROLLARY 6.15. Let , g Pn, r . Then q Ž a . g R, and by the specialization 0 , q Ž a . is mapped to 1 Ž w .. Proof. It follows from Ž6.11.2. that q˜Ž x; q, u. is an R-linear combination of various m Ž x ., and so is an R-linear combination of S Ž x .. Since Schur functions S Ž x . are linearly independent for g Pn, r , Theorem 6.14 implies that q Ž a . g R. Moreover, the argument used to prove Theorem 6.14 works well for W, and we have q˜ Ž x ; 1, . s
Ý
g Pn, r
1 Ž w . S Ž x . ,
Ž 6.15.1.
where stands for the substitution of , 2 , . . . , r s 1 into u1 , u 2 , . . . , u r . ŽHere q˜Ž x; 1, . coincides with PŽ x ... Hence the second assertion follows by specializing the equation Ž6.14.1. by 0 , and by comparing it with Ž6.15.1..
7. CHARACTER VALUES 7.1. Let l: W ª ⺪ G 0 be the length function of W with respect to the generators t 1 , s2 , . . . , sn4 ; i.e., l Ž w . is the smallest number of generators which are needed to express w as the product of such generators. The function l Ž w . is determined in Bremke and Malle wBMx. Some properties of l Ž w . were studied further in wRSx. In particular, the following is known Žcf. wRS, 1.31x.. For each k Ž1 F k F n. and a Ž0 F a F r y 1., put
n Ž k, a . s
½
skq 1 ⭈⭈⭈ sny1 sn sk ⭈⭈⭈ s2 t a s2 ⭈⭈⭈ sn
if a s 0, if a / 0.
Ž 7.1.1.
Then the set Rn s nŽ k, a. ¬ 1 F i F n, 0 F a F r y 14 gives rise to a complete set of representatives of the left cosets Wn, rrWny1, r . Moreover, we have l Ž ww⬘. s l Ž w . q l Ž w⬘. for w g Rn , w⬘ g Wny1, r . The expression of nŽ k, a. in Ž7.1.1. is a reduced expression with respect to l. The decomposition for Wny 1, r _ Wn, r is also described in wRS, Corollary 1.29x. Combining with the previous result, we have the following double coset decomposition of Wn, r by Wny1, r . Ž7.1.2. Let R˜n s sn , t na ¬ 0 F a F r y 14 . Then any element w g Wn, r can be written as w s w⬘ dw⬙ with w⬘, w⬙ g Wny 1, r , d g R˜n , such that l Ž w . s l Ž w⬘. q l Ž d . q l Ž w⬙ .. Let C s C be the conjugacy class in W corresponding to g Pn, r . We denote by Cmin the set of w g C such that l Ž w . is minimal in C. The following lemma is related to Lemma 2.5 in Geck and Pfeiffer wGPx.
850 LEMMA 7.2.
TOSHIAKI SHOJI
Let C s C for g Pn, r . Then we ha¨ e Cmin l ⌺ / ⭋.
Proof. The proof is done by using a similar argument as in wGP, Proposition 2.3x. Let us take w g C. One can write w s xd n y by Ž7.1.2., with x, y g Wny 1, r and d n g R˜n . Let w⬘ s xy1 wx s d n yx. Then l Ž w⬘. F l Ž w . by the property of the length function in Ž7.12.. Again by Ž7.12., applied to Wny 1, r , we have yx s x⬘d ny1 y⬘ with x⬘, y⬘ g Wny2, r and d ny1 g R˜ny 1. Since elements in R˜n commute with Wny2, r , we see that d n yx s d n x⬘d ny1 y⬘ s x⬘d n d ny1 y⬘. Hence, by a similar argument as before, w⬙ s d n d ny1 y⬘ x⬘ is in C and l Ž w⬙ . F l Ž w .. Repeating this procedure, we can find w 1 s d n d ny1 ⭈⭈⭈ d1 d 0 g C such that l Ž w 1 . F l Ž w ., where d i g R˜i for i G 1 and d 0 g t 1a ¬ 0 F a F r y 14 . The lemma follows from this since w 1 is contained in ⌺ . Remark 7.3. The proof of Theorem 6.14 implies that q Ž a w . takes a constant value for w g ⌺ . Hence q Ž a w . is constant for w g Cmin l ⌺ . In comparing the case of Iwahori᎐Hecke algebras Žsee 7.4 below., it seems interesting to know whether q Ž a w . is constant for w g Cmin . An alternative length function, nŽ w . in the notation in wRSx, associated to the root system of W was introduced in wBMx. One can define the set X Cmin of minimal elements with respect to nŽ w .. It would be also interesting X to consider a similar problem as above for Cmin . 7.4. In wGPx Geck and Pfeiffer have shown, extending the result of wRx for the Hecke algebra of type A n , that in the case of Iwahori᎐Hecke algebras, any character has a constant value on the standard basis Tw for w g Cmin , where Cmin is the set of minimal length element in a given conjugacy class C of the Weyl group, and that the character values at other Tw ⬘ are determined from the values at Tw with w g Cmin . The following result, which is a generalization of Theorem 5.1 in wRx, together with Lemma 7.2, is, in a sense, a counterpart of their result for the case of complex reflection groups GŽ r, 1, n., although our basis of HK is different from the standard basis. This answers, in some sense, a question posed in the last paragraph in Pfeiffer wPx. PROPOSITION 7.5.
For each w g W, there exists Aw s
Ý
g Pn, r
c w , a
with c w, g R1 , such that q Ž a w . s q Ž A w . for any character q of H . In particular, the characters q are determined by a with g Pn, r .
ARIKI ᎐ KOIKE ALGEBRAS
851
Proof. Let ˆ ⌺ be the subset of W consisting of w s t 1c1 ⭈⭈⭈ t nc n such that any k-cycle 1 occurring in the decomposition of is of the form 1 s s pqky2 ⭈⭈⭈ s pq1 s p for some p G 2. Here we introduce a notation. For any w s t 1c1 ⭈⭈⭈ t nc n , we define l 1Ž w . by l 1Ž w . s l Ž .. First we show the following. Ž7.5.1. For each w g W, there exists AXw s
Ý
cw , z a z g H h
ˆ zg⌺
with c w, z g R1 , such that q Ž a w . s q Ž AXw . for any character q of H . Moreover, l 1Ž z . F l 1Ž w . if c w, z / 0. Now take w s t 1c1 ⭈⭈⭈ t nc n g W, and let i be the largest integer such that Ž i . - i y 1. We shall show the existence of AXw by induction on i and reverse induction on Ž i .. Note that if such an i does not exist, then w is contained in ˆ ⌺. Moreover, if i is such a number with Ž i . s n, then we may replace 1, 2, . . . , n4 by 1, 2, . . . , i y 14 . So, let i be such a number with Ž i . / n, and put j s Ž i . q 1. Then j - i and so y1 Ž j . - i. It follows that l Ž s j . - l Ž .. Put ⬘ s s j s j and ⬙ s s j . First assume that l Ž ⬘. ) l Ž ⬙ .. Then a s a j a ⬙ . We have
q Ž a w . s q Ž 1c1 ⭈⭈⭈ nc n a j a ⬙ . s q Ž a j f 1 Ž . a ⬙ . q q Ž f 2 Ž . a ⬙ . , by Ž3.6.6., where f 1 , f 2 are polynomials in 1 , . . . , n with coefficients in R1. Now
q Ž a j f 1 Ž . a ⬙ . s q Ž f 1 Ž . a ⬙ a j . s q Ž f 1 Ž . a ⬘ . . But we have ⬘Ž i . s j s Ž i . q 1, and also ⬙ Ž i . s j. Hence the induction hypothesis works. Next assume that l Ž ⬘. - l Ž ⬙ .. Then a ⬙ s a ⬘ a j and a s a j a ⬘ a j . Hence
q Ž a w . s q Ž 1c1 ⭈⭈⭈ nc n a j a ⬘ a j . s q Ž a j f 1 Ž . a ⬘ a j . q q Ž f 2 Ž . a ⬘ a j . . But since
q Ž a j f 1 Ž . a ⬘ a j . s q Ž f 1 Ž . a ⬘ a2j . s Ž q y qy1 . q Ž f 1 Ž . a ⬙ . q q Ž f 1 Ž . a ⬘ . ,
852
TOSHIAKI SHOJI
we have
q Ž a w . s q Ž f 1 Ž . a ⬘ . q q Ž f 3 Ž . a ⬙ . for certain polynomials f 1 , f 3 . Since ⬘Ž i . s ⬙ Ž i . s j s Ž i . q 1, the induction hypothesis works. The latter assertion is easily checked for both cases. Thus we get Ž7.5.1.. We now prove the proposition by induction on l 1Ž w .. By Ž7.5.1., we may assume that w g ˆ ⌺. Then w is a product of the elements of the form c ly 1 c l t ly1 t l ⭈⭈⭈ t kc k sk sky1 ⭈⭈⭈ sl for some integers l F k. Accordingly, a w is a c ly 1 c l product of Žmutually commuting. elements of the form ly1 l ⭈⭈⭈ ck k a k a ky1 ⭈⭈⭈ a l . We consider the special case where w s t 1c1 t 2c 2 ⭈⭈⭈ t nc n sn sny1 ⭈⭈⭈ s2 . Put s sn ⭈⭈⭈ s2 . Then c ny 2 c n c ny1 q Ž a w . s q Ž 1c1 ⭈⭈⭈ ny2 n a n a ny1 ⭈⭈⭈ a2 ny1 . c ny 2 qc ny 1 c n s q Ž 1c1 ⭈⭈⭈ ny2 n a n a ny1 ⭈⭈⭈ a2 . q
Ý
⬘-
q Ž f ⬘ Ž . a ⬘ .
by Ž3.6.5. ᎐ Ž3.6.8., with some polynomials f ⬘ on . Then by repeating this procedure for ny 2 , ny3 , . . . , 1 , we have
q Ž a w . s q Ž nc1q ⭈⭈⭈ qc n a n a ny1 ⭈⭈⭈ a2 . q
Ý
⬘-
q Ž fX ⬘ Ž . a ⬘ .
fX ⬘
for some polynomials on . Now z s nc1q ⭈⭈⭈ qc n a n ⭈⭈⭈ a2 g ⌺ for some g Pn, r . On the other hand, Proposition 6.10 and Theorem 6.14 imply Žby the linear independence of Schur functions. that q Ž a z . s q Ž a .. Hence the proposition holds in this case. The argument for the general case is completely similar. Remark 7.6. Proposition 7.5 and Corollary 6.15 imply that q Ž a w . g R1. But this also follows from Proposition 4.3. More precisely, we see that q Ž a w . is mapped to 1 Ž w . by the specialization 0 . In fact, by Proposition 4.3, any irreducible representation Z has an R1 form Žsince an irreducible Hn-module is realized on R1 .. Moreover, by the specialization 0 this turns out to be an irreducible W-module Z . The above assertions follow easily from this. 7.7. Let Wqs Wq Ž . be as in 5.1, and let 1 be the linear character of Wq as given in 5.5. We consider the induced character IndW W q 1 of W, which will be denoted by ⌰ 1 . The value of ⌰ 1 at w is interpreted in terms of PŽ x . and m Ž x . as follows Žrecall that PŽ x . is the complex conjugate of PŽ x ..: P Ž x . s
Ý
g Pn, r
⌰1 Ž w . m Ž x . .
Ž 7.7.1.
ARIKI ᎐ KOIKE ALGEBRAS
853
In fact, if we write P s Ý c , m , the coefficient c , is described as follows. Let G be the set of maps from the set X s Ž i, j . ¬ 1 F i F r, 1 F j F l Ž Ž i. .4 to 1, 2, . . . , r 4 . Then g g G determines a multipartition Ž g . s Ž Ž1., . . . , Ž r . . g Pn, r , where Ž k . is obtained from the set Žj i. ¬ g Ž i, j . s k 4 by arranging the order. Let F Ž g, i, . be the set of maps f from the set 1, 2, . . . , l Ž Ž i. .4 to 1, 2, . . . , r 4 such that Ý j jŽ i. s Žki. Žwhere the sum is taken over all j such that f Ž j . s k .. Then, it follows from the definition of P and m that we can write r
c , s
Ý
Ł
gg G Ž i , j .gX
i⭈ g Ž i , j. Ł F Ž g , k , . .
Ž 7.7.2.
ks1
On the other hand, it is easy to compute the value ⌰ 1 Ž w . explicitly, which coincides with the value c , in Ž7.7.2.. Thus Ž7.7.1. holds. For each partition s Ž 1 , 2 , . . . . of n, we set zŽ t . s z ⭈ Ł i G 1Ž1 y i .y1 t , where z is the order of the centralizer of in ᑭ n . For a multipartition s Ž Ž1., . . . , Ž r . ., we define a function z Ž t . by r
z Ž t . s r lŽ . Ł zŽ i. Ž t . .
Ž 7.7.3.
is1
For a partition s Ž 1 , 2 , . . . , k ., the Hall᎐Littlewood function k qŽ x; t . is defined as qŽ x; t . s Ł is1 q iŽ x; t .. For a multipartition g Pn, r , we define a function q Ž x; t . by r
qŽ x ; t . s
Ł q Ž x Ž i. ; t . .
Ž 7.7.4.
Ž i.
is1
We now introduce infinitely many variables, x iŽ k ., yiŽ k ., for i s 1, 2, . . . , and 1 F k F r. We have the following lemma. LEMMA 7.8.
Let r
⍀ Ž x, y ; t . s
ŁŁ
ks1 i , j
1 y tx iŽ k . y jŽ k . 1 y x iŽ k . y jŽ k .
.
Then we ha¨ e ⍀ Ž x, y ; t . s
Ý qŽ x ; t . m Ž y . ,
Ž 7.8.1.
Ý z Ž t . y1 P Ž x . P Ž y . ,
Ž 7.8.2.
⍀ Ž x, y ; t . s
where in both formulas, runs o¨ er the multipartitions s Ž Ž1., . . . , Ž r . . of any size.
854
TOSHIAKI SHOJI
Proof. The equality Ž7.8.1. follows easily from the equality for the case where r s 1, Žcf. wM, III, Ž4.2.x.. The equality Ž7.8.2. for r s 1 is given in wM, III, Ž4.1.x. By a similar argument as in the case r s 1, we have log ⍀ Ž x, y ; t . s
⬁
1 y tm
ÝÝ Ý k
m
i , j ms1
Ž x iŽ k . yjŽ k . .
m
.
But since r
1
Ý Ž yk k ⬘ .
r
as1
r
⬁
a
s ␦k , k⬘ ,
we have log ⍀ Ž x, y ; t . s
Ý Ý
1 y tm
Ý
as1 ms1 k , k ⬘, i , j ⬁
r
s
Ý Ý as1 ms1
1 y tm rm
rm
yak ak ⬘ Ž x iŽ k . y jŽ k ⬘. .
m
PmŽ a. Ž x . PmŽ a. Ž y . .
Hence r
⍀ Ž x, y ; t . s
⬁
Ł Ł exp
as1 ms1
s
½
1 y tm rm
PmŽ a. Ž x . PmŽ a. Ž y .
5
Ý z Ž t . y1 P Ž x . P Ž y . ,
as asserted. Thus, the lemma is proved. 7.9. Returning to the Hecke algebra setting, we shall express the character value q Ž a . in terms of the characters 1 and ⌰1 Žfor some . of W. For this, we introduce some notation. For a partition s Ž 1 , 2 , . . . , k ., we denote by M the set of k = r matrices C s Ž c i j . with integers c i j G 0 such that Ý rjs1 c i j s i . For C g k M, we define an element u C s Ł is1 u nŽ i. , where nŽ i . is the largest j such that c i j / 0 Žfor a fixed i .. Also we associate s Ž C . s Ž Ž1., . . . , Ž r . . g Pn, r to C g M as follows: the partition Ž j. is obtained from the set c i j ¬ 1 F i F k 4 by arranging the order. More generally, for a multipartition g Pn, r , we consider the set M s MŽ1. = ⭈⭈⭈ = MŽ r . , and for each C s Ž C1 , . . . , Cr . g M , we put u C s r Ł is1 u Ci i . The multipartition s ŽC. s Ž Ž1., . . . , Ž r . . associated to C g M is defined as the union of Ž Ci .; i.e., Ž j. is the union of corresponding Ž jth. partitions in Ž Ci ..
ARIKI ᎐ KOIKE ALGEBRAS
855
Now by making use of Proposition 6.12, q˜Ž x; q, t . may be interpreted by the following formula: q˜ Ž x ; q, u . s
qn
Ž q y qy1 .
lŽ .
Ý
u C q ŽC. Ž x ; qy2 . .
Ž 7.9.1.
Cg M
We have the following result. THEOREM 7.10.
q Ž a . s
For each , g Pn, r , we ha¨ e qn y1 l Ž .
Žq y q .
Ý
uC
Ý
g Pn , r
Cg M
z Ž qy2 .
y1
1 Ž w . ⌰1 ŽC. Ž w . .
Proof. The proof is similar to wR, Theorem 5.4x. First we compute the coefficient b , of S Ž x . in the expansion of q Ž x; t . in terms of Schur functions. Then b , coincides with the coefficient of S Ž x . m Ž y . in ⍀ Ž x, y; t . by Ž7.8.1.. Hence by Ž7.8.2., it is equal to the coefficient of S Ž x . m Ž y . in Ý z Ž t .y1 P Ž x . P Ž y .. But Ž6.1.4. implies that P Ž x . s
Ý 1 Ž w . S Ž x . .
It follows, by virtue of Ž7.7.1., that b , s
Ý z Ž t . y1 1 Ž w . ⌰1 Ž w . .
Ž 7.10.1.
Now by the Frobenius formula ŽTheorem 6.14., q Ž a . coincides with the coefficient of S Ž x . in q˜Ž x; q, u.. Hence the theorem follows from Ž7.9.1. together with Ž7.10.1.. Remark 7.11. Theorem 7.10 expresses the character value q Ž a . in terms of the values of irreducible characters and certain induced characters of W. The corresponding formula for Hn was given in wR, Theorem 5.4x. Our proof of Theorem 7.10 is a generalization of his argument. However, essentially the same result as Ram had already been obtained by Starkey wSx in his Ph.D. thesis in 1975, though it was not published at all. The author is grateful to M. Geck for communicating him the result of Starkey. See wGex for more details on Starkey’s work and related topics.
REFERENCES wAKx S. Ariki and K. Koike, A Hecke algebra of Ž⺪rr⺪. X ᑭ n and construction of its irreducible representations, Ad¨ . Math. 106 Ž1994., 216᎐243.
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TOSHIAKI SHOJI
wBMx K. Bremke and G. Malle, Reduced words and a length function for GŽ e, 1, n., Indag. Math. 8 Ž1997., 453᎐469. wGex M. Geck, The character table of the Iwahori᎐Hecke algebra of the symmetric group: Starkey’s rule, C. R. Acad. Sci. Paris Ser. ´ I. Math. 329 Ž1999., 361᎐366. wGPx M. Geck and G. Pfeiffer, On the irreducible characters of Hecke algebras, Ad¨ . Math. 102 Ž1993., 79᎐94. wGx A. Gyoja, A q-analogue of Young symmetrizer, Osaka J. Math. 23 Ž1986., 841᎐852. wJx M. Jimbo, A q-analogue of UŽ ᒄ ᒉ Ž N q 1.., Hecke algebra, and the Yang᎐Baxter equation, Lett. Math. Phys. 11 Ž1986., 247᎐252. wMx I. G. Macdonald, ‘‘Symmetric Functions and Hall Polynomials,’’ 2nd ed., Clarendon, Oxford, 1995. wPx G. Pfeiffer, Character values of Iwahori᎐Hecke algebras of type B, in ‘‘Finite Reductive Groups, Related Structures and Representations ŽM. Cabanese, Ed.., Progress in Math. Vol. 141, pp. 333᎐360, Birkhauser, Boston, 1997. ¨ wRx A. Ram, A Frobenius formula for the characters of the Hecke algebras, In¨ ent. Math. 106 Ž1991., 461᎐488. wRSx K. Rampetas and T. Shoji, Length functions and Demazure operators for GŽ e, 1, n., I, Indag. Math. 9 Ž1998., 563᎐580. wSSx M. Sakamoto and T. Shoji, Schur᎐Weyl reciprocity for Ariki᎐Koike algebras, J. Algebra 221 Ž1999., 293᎐314. wSx A. J. Starkey, ‘‘Characters of the Generic Hecke Algebra of a System of BN-pairs,’’ Ph.D. thesis, University of Warwick, 1975. wWx H. Wenz, Hecke algebras of type A n and subfactors, In¨ ent. Math. 92 Ž1988., 349᎐383.