- Email: [email protected]
Mod-pReduction for Quantum Groups
202, 357]366 Ž1998. JA977240
JOURNAL OF ALGEBRA ARTICLE NO.
Mod-p Reduction for Quantum Groups N. Cantarini Dipartimento di Matematica, Uni¨ ersita ` di Pisa, Via Buonarroti 2, Pisa, Italy Communicated by Corrado de Concini Received June 22, 1997
Let Ue Ž G . be the quantized enveloping algebra associated to the Lie algebra G s sl Ž n q 1. at a pth-root of unity « and assume that p is a prime which does not divide n q 1. It is known that the irreducible, finite dimensional representations of U« Ž G . are parametrized, up to isomorphisms, by the conjugacy classes of SLŽ n q 1.. In the paper we prove that the dimension of any U« Ž G .-module M parametrized by a conjugacy class O is divided by p1r 2 dimŽ O .. This result was conjectured by C. De Concini, V. G. Kac, and C. Procesi Ž J. Amer. Math. Soc. 5, 1992, 151]190.. Q 1998 Academic Press
INTRODUCTION Let Uq Ž G . be the quantized enveloping algebra associated to a simple, finite dimensional Lie algebra G according to the definition given by Drinfeld wD2x and Jimbo wJx, and let U« Ž G . be the algebra over C obtained from Uq Ž G . by specializing q at a primitive lth-root of unity « , l being a fixed odd integer strictly greater than 1. In the papers wDC-K1, DC-K-P1x it is shown how to parametrize the irreducible, finite dimensional representations of U« Ž G . by the conjugacy classes of the algebraic group G with Lie algebra G and trivial center. In particular it was conjectured that the dimension of any irreducible, finite dimensional U« Ž G .-module, corresponding to a conjugacy class O , is divisible by l 1r2 dimŽ O .. The main purpose of this paper is to prove this conjecture for the irreducible representations of the quantized enveloping algebra associated to G s sl Ž n q 1.. We recall that the regular case, that is, the case of the representations corresponding to a conjugacy class of maximal dimension, was proved by 357 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
358
N. CANTARINI
De Concini, Kac, and Procesi in wDC-K-P2x. According to wDC-K2x, any irreducible representation of U« Ž sl Ž n q 1.. is either unipotent Žsee Section 2 for the definition. or induced by a unipotent one. Therefore one can reduce to the study of the unipotent representations ŽSection 2.. The above conjecture is the quantum-analogue of a well known conjecture which was formulated by Kac and Weisfeiler in wK-W1x and proved by Premet in wPx. It states that if G is a simple and simply connected algebraic group defined over an algebraically closed field, whose characteristic p ) 0 is good for G Žsee Section 3. and such that G s LieŽ G . admits a non-degenerate G-invariant trace form, then any irreducible G-module V has dimension divisible by p1r2 dimŽ V Ž x .. , where x g G U is the p-character of V and V Ž x . is its coadjoint orbit in G U . We recall that, also for the Kac]Weisfeiler conjecture, a theorem of reduction to the nilpotent case, similar to the theorem mentioned above, had been proved in wK-W1x. We make use of Premet’s theorem to obtain our result. More precisely, we fix a prime p and a primitive pth-root of unity « and, starting from an irreducible U« Ž sl Ž n q 1..-module M parametrized by a unipotent conjugacy class O of SLŽ n q 1., we construct a representation M of the Lie algebra sl Ž n q 1. over an algebraically closed field of characteristic p, such that dimŽ M . s dimŽ M . and the dimension of O is equal to the dimension of the coadjoint orbit of the p-character of M. In order to realize this construction we will be considering the ‘‘limiting’’ specialization U1 of Uq Ž G . ŽSection 1.. Besides, taking into account Premet’s results wPx, we shall assume p ¦ n q 1. One fundamental step in our construction consists in proving that any irreducible, finite dimensional, unipotent U« Ž sl Ž n q 1..-module is a quotient of an induced module ŽSection 3.. I thank Professor Corrado De Concini for his precious suggestions and the long discussions we had.
1. DEFINITIONS AND NOTATIONS Let Ž a i j . i, js1, . . . , n be a symmetric Cartan matrix and let G be the associated simple, finite dimensional Lie algebra. By Q and Qq we shall denote the sets of roots and positive roots, respectively, with N s < Qq< , by R the root lattice, and by D a fixed set of simple roots. DEFINITION 1.1. The quantized enveloping algebra Uq Ž G . associated to Ž a i j . i,n js1 is the associative algebra over CŽ q . with generators Ei , Fi , K i" 1,
MOD-p REDUCTION FOR QUANTUM GROUPS
359
for i s 1, . . . , n, and the following defining relations: Ei Fj y Fj Ei s d i j K i K j s K j K i s K iqj , K i Ej Ky1 s q a i j Ej , i 1ya i j s Ý Ž y1.
ss0 1ya i j s Ý Ž y1.
ss0
K i y Ky1 i
Ž 1.1.
q y qy1
s Ky1 K i Ky1 i i Ki s 1
Ž 1.2.
K i Fj Ky1 s qya i j Fj i
Ž 1.3.
1 y a i j 1y a i jys Ei Ej Eis s 0 s
if i / j
Ž 1.4.
1 y a i j 1y a i jys Fi Fj Fi s s 0 s
if i / j.
Ž 1.5.
Here 1 ys a i j is the Gaussian binomial coefficient for example, wDC-K1x..
1 y ai j s d
with d s 1 Žsee,
Let W be the Weyl group associated to Ž a i j . and w 0 g W its element with maximal length: l Ž w 0 . s N. By TW we denote the corresponding Braid group whose generators Ti , with i s 1, . . . , n, act on Uq Ž G . according to wL2x. It is known that if we fix a reduced expression of w 0 : w 0 s si1 si 2 ??? si N , then we can define a total convex ordering in Qq by putting
b j s si1 ??? si jy 1Ž a i j . for j s 1, . . . , N. We define Eb j s Ti1 ??? Ti jy 1 Ei j ,
Fb j s Ti1 ??? Ti jy 1 Fi j
N and, for every k s Ž k 1 , . . . , k N . g Zq ,
E k s Ebk11 ??? EbkNN ,
F k s FbkNN ??? Fbk11 .
Let us fix a prime p and a primitive pth-root of unity « . By U« Ž G . we denote the associative algebra over C with generators Ei , Fi , K i" 1 and relations Ž1.1. ] Ž1.5. where q is replaced by « . Now put A s Zw q, qy1 x and take the A-subalgebra U A of Uq Ž G . generated by the elements Ei , Fi , K i , and Hi [ w Ei , Fi x, with the relations Ž1.1. ] Ž1.5. and K i y Ky1 s Ž q y qy1 . Hi . We define i U1 [ UAr Ž f qp , q y 1 . UA , where f qp is the pth cyclotomic polynomial in the variable q.
360
N. CANTARINI
Notice that the following isomorphism of fields holds: Z q, qy1 r Ž f qp , q y 1 . , ZrpZ. Furthermore we recall the following well known result: PROPOSITION 1.2 wL1x. U1 , UZr pZ Ž G . m ZrpZwŽZr2Z. r k G x where UZr pZ Ž G . is the uni¨ ersal en¨ eloping algebra of G o¨ er ZrpZ. 2. REDUCTION TO THE EXCEPTIONAL CASE Let RepŽ U« . be the set of the equivalence classes of the irreducible, finite dimensional representations of U« . Let Z« be the center of U« . Then Žsee wDC-K1x. the elements Eap , Fap , with a g Qq, and K ip , with i s 1, . . . , n, lie in Z« . Besides, if Z0 is the subalgebra of Z« generated by these elements, then U« is a free Z0-module with a basis consisting of the N vectors F k K 1m 1 ??? K nm n E r < k s Ž k 1 , . . . , k N ., r s Ž r 1 , . . . , rN . g Zq , mj g Z, 0 F k j , r j , m j F p y 14 . By G we shall denote the algebraic group whose Lie algebra is G and whose center is trivial. It is known wDC-K-P1x that the elements of RepŽ U« . are parametrized by the conjugacy classes of the group G as it is stated by the following theorem: THEOREM 2.1 wDC-K-P1x. There exists a map
w : Rep Ž U« . ª G such that: Ža. ImŽ w . is the big cell of G; Žb. the representations corresponding to conjugated elements in G are all of the same ‘‘type’’ Ž i.e., they are equi¨ alent, up to a twist, by the elements ˜ of automorphisms of U« ., in particular they ha¨ e the same of a group G dimension. In fact it has been shown by De Concini and Kac wDC-K2x that, in order to classify the irreducible, finite dimensional representations of U« , it is sufficient to consider the representations corresponding to the conjugacy classes of some special elements called exceptional. DEFINITION 2.2. A semisimple element g in G is called exceptional if the center of its centralizer ZG Ž g . in G is finite. DEFINITION 2.3. An element g in G is called exceptional if its semisimple part, with respect to the Jordan decomposition, is exceptional.
MOD-p REDUCTION FOR QUANTUM GROUPS
361
DEFINITION 2.4. A representation s in RepŽ U« . is called exceptional if w Ž s . is an exceptional element of G. Let us now take a non-exceptional representation s in RepŽ U« . on a vector space V, with central character x . Then we may regard s as a representation of the algebra Ux [ U«rŽ z y x Ž z ., z g Z0 .. As g s w Ž s . ˜ is not exceptional, we may suppose, up to the action of some element of G on s , that if g s g s g u is the Jordan decomposition of g, then g satisfies the following conditions: }g u belongs to the maximal unipotent subgroup Ny of G, corresponding to yQq; }h G [ LieŽcenterŽ ZG Ž g s ... / 0; }QX [ a g Q < a vanishes on h G 4 s ZDX l Q where ZDX is a sublattice of R spanned by a proper subset DX of D. Let U«X be the subalgebra of U« generated by the K i ’s, Ky1 i ’s, with i s 1, . . . , n and by the elements Ej , Fj such that a j g DX . Put UxX s U«Xr Ž z y x Ž z . , z g Z0 l U«X . , U g s U«X Uq,
Uxg s U gr Ž z y x Ž z . , z g Z0 l U g . ,
where Uq is the subalgebra of U« generated by the Ei ’s. Then the following theorem holds: THEOREM 2.5 wDC-K2x. There exists a unique irreducible Uxg-submodule V of V such that: X
Ža. V X is a UxX-module. Žb. The Ux-module V is induced by V X , V s Ux mUxg V X . In particular dimŽ V . s p t dimŽ V X ., where 2 t s < Q _ QX <. Žc. The map V ¬ V X is a bijection RepŽ U« . ª RepŽ UxX .. Remark. If we restrict the representation of UxX in V X to the subalgebra of UxX generated by the elements Ei , Fi , K i" 1 such that a i g DX , we get an irreducible representation. In fact this is an exceptional representation of the quantum group U« Ž G X . where G X is the subalgebra of G generated by its Chevalley generators corresponding to a i g DX . We recall that if G s sl Ž n q 1. the exceptional elements are the unipotent ones. Furthermore every unipotent matrix in SLŽ n. is conjugated to
362
N. CANTARINI
one with the form Jh 1 Ls
0
Jh 2
.. 0
. Jh t
0
,
Ž 2.6.
where h1 G h 2 G ??? G h t is a partition of n and Jt is a t = t Jordan block: 1 Jt s
^
1 .. .
0
..
. ..
0
.
`
1 1
0
.
_
t
We underline that this classification does not depend on the characteristic of the field.
3. THE A n-CASE In this section we shall examine the case G s sl Ž n q 1.. We know that the Weyl group W is the symmetric group Snq 1. From now on we shall assume that a reduced expression of w 0 , say w 0 s s n s ny 1 ??? s1 s n ??? s 2 s n ??? s n , and the associated ordering b 1 , . . . , bN 4 of the positive roots have been fixed. Let O be a conjugacy class of SLŽ n q 1. parametrized by a unipotent matrix L of type Ž2.6. for some blocks Jh1, . . . , Jh t , with h1 G h 2 G ??? G h t a partition of n q 1. Take an irreducible U« Ž G .-module M such that w Ž M . is conjugated to L. Our first aim is to show that every such a module is the quotient of an induced module. According to the definition of w Žsee wDC-K-P1x., for every positive root a we have Eap s 0, Fap either 0 or 1, and for every i s 1, . . . , n, K ip s 1 Žby Ea , Fa , K j we denote both the generators of sl« Ž n q 1. and their images through the representation .. We consider the set B s x g M < Ei Ž x . s 0 ; i s 1, . . . , n4 .
MOD-p REDUCTION FOR QUANTUM GROUPS
363
Since the subalgebra of sl« Ž n q 1. generated by the Ei ’s acts nilpotently, B is a non-trivial set upon which the K i ’s act diagonally; thus we can take a non-zero element m g B such that K i m s « s i m with 0 F si F p y 1. Now, if U U is the C-subalgebra of U« Ž sl Ž n q 1.. generated by Ei , K i , Fap for every i s 1, . . . , n, a g Qq, then the line L generated by m is an U Umodule. Since M is irreducible it is generated by m hence M is a quotient of the induced module I [ U« Ž sl Ž n q 1 . . mU U L. We have thus established the following proposition: PROPOSITION 3.1. Let M be an irreducible, finite dimensional U« Ž sl Ž n q 1..-module lying o¨ er a unipotent canonical element of SLŽ n q 1.. Then there exist a proper subalgebra U U of U« and an element m g M _ 04 such that, if L is the line generated by m, then: Ž1. Ž2. of I.
L is an U U -module; M s IrS where I [ U« Ž sl Ž n q 1.. mU U L and S is a submodule
Remark. The induced module I has got a natural basis C : C s Fbh11 ??? FbhNN m < 0 F h j - p 4 . If C s Zw « x-span C 4 then C is stable with respect to the action of the Ei ’s, Fi ’s, K i ’s, Hi ’s for every i s 1, . . . , n. Indeed, for every z in C such that z K j z s « l j z Ž0 F l jz F p y 1. the following relations hold: Ej Fjk z s Fjk Ej z q Ž1 y« 2 k . Ž « 2y2 kq l j y «y l j .rŽ1 y« 2 .Ž « y«y1 . z
v
Fjky 1 z;
z
1 k Ej Fhk. . . j z s yŽ1 y « 2 k .rŽ1 y « 2 . «y l j Fhky . . . j Fh . . . jy1 z q Fh . . . j Ej z if z
v
h - j;
k Ej Fjk. . . t z s Ž1 y «y2 k .rŽ1 y «y2 . « l j q1 Fjky1 . . . t Fjq1 . . . t z q Fj . . . t E j z if z
v
t ) j; v v v v v
Ej Fhk. . . t s Fhk. . . t Ej if h - j - t; Fj Fhk. . . t s Fhk. . . t Fj if h - j - t; Fj Fjk. . . t s « k Fjk. . . t Fj if t ) j; Fj Fhk. . . j s «yk Fhk. . . j Fj if h - j; k yk k ky1 Fj Fjq1 Fjq1 . . . t Fj q «yk Ž1 y « 2 k .rŽ1 y « 2 . Fjq1 ... t s « . . . t Fj . . . t .
We notice that, since U« Ž sl Ž n q 1.. is a finitely generated algebra, the module M is defined over a finite extension K of QŽ « . and, in order to compute the dimension of M, we may reduce to K, instead of C.
364
N. CANTARINI
Let L be the localization of the ring of integers Zw « x of QŽ « . at the ideal P0 generated by « y 1, by m P 0 denote the maximal ideal of L , and let U L s UArf qp UA mZw « x L . Then the induced module I is defined over U L . Let us choose an ideal P, in the ring of integers Z of K, lying over P0 and let L X be the localization of Z at P, m P its maximal ideal. Then L Xrm P is a finite extension of Lrm P 0 and thus a finite field of characteristic p. We put U L X s U L mL L X . PROPOSITION 3.2. Let M be an irreducible U« Ž sl Ž n q 1..-module such that w Ž M . is a unipotent canonical element in SLŽ n q 1.. Then there exist a finite extension K of QŽ « ., a local ring L X in K, and an U L X-module M X such that M X is a free L X-module and M s M X mL X K . Proof. We use the same notations as in Proposition 3.1; then we know that M is the quotient of the induced module I by a submodule S. According to the remark on the basis C of I, we have I s I P mL X K , where I P is the module, over the localization L X of Z in P, naturally defined. Now, since L X is a discrete valuation ring, S l I P is free and it is a direct factor of I P , hence, if we put M X s I P rŽ S l I P . we obtain the result. In particular, rkŽ S l I P . s dim K S and dim K M s rkŽ M X .. Notice that the action of U« on M is obtained by extending that of UL X . Before coming to a conclusion we recall the following theorem: THEOREM 3.3 wPx. Let G be a simple and simply connected algebraic group defined o¨ er an algebraically closed field whose characteristic p ) 0 is good for G. Suppose that G s LieŽ G . admits a non-degenerate G-in¨ ariant trace form. Let V be an irreducible G-module with p-character x g G U . Then the dimension of V is di¨ isible by p1r2 dimŽ V Ž x .. , where V Ž x . is the coadjoint orbit of x . Here by a ‘‘good’’ prime we mean a prime greater than any coefficient of any positive root in its expression relative to a basis of simple roots. Therefore if G s sl Ž n q 1. any prime p is good for G . On the other hand, in order to guarantee the existence of a non-degenerate trace form on G , we shall suppose p ¦ n q 1. In fact in wPx it is shown that the
MOD-p REDUCTION FOR QUANTUM GROUPS
365
Kac]Weisfeiler conjecture is no longer true for G s sl Ž mp, k. when m G 1 Ž p is the characteristic of the field k.. We are now ready to state our main result: THEOREM 3.4. Let « be a primiti¨ e pth-root of unity with p prime, p ) 2, p ¦ n q 1. If M is an irreducible, finite dimensional U« Ž sl Ž n q 1..-module lying o¨ er a conjugacy class O , then dimŽ M . s kp1r2 dimŽ O . for some positi¨ e integer k. Proof. As we explained in Section 2 we may reduce to the unipotent case and thus suppose that O is the conjugacy class of a unipotent element of SLŽ n q 1.. Then let us assume the construction of the module M X and, ˜ s M Xrm P M X . As L X is a using the same notations as above, let us put M X ˜ .. Besides, by its own definition, M˜ is a local ring, rkŽ M . s dim L X r m PŽ M U1-module and therefore a UZr pZ Ž G .-module Žsee Section 1.. We finally denote the algebraic closure of the finite field L Xrm P by L Xrm P and define
˜ mL X r m P L Xrm P . MsM If x is the p-character of M, then dim K Ž M . s dim L Xrm P Ž M . s kp1r2 dimŽ V Ž x .. for some positive integer k. Indeed, since M is finite-dimensional, it has a composition series consisting of irreducible modules M j satisfying the hypotheses of Theorem 3.3, whose dimension is therefore divided by p1r2 dimŽ V Ž x ... We point out that the canonical unipotent element L of type Ž2.6., corresponding to the Jordan partition l, reduces ‘‘modulo p’’ to the canonical nilpotent element corresponding to l, that is, to the nilpotent element with the same blocks as L but with all zeros on the diagonal. Therefore the dimension of the coadjoint orbit V Ž x . of x is the dimension of the conjugacy class O .
REFERENCES wDC-K1x wDC-K2x wDC-K-P1x wDC-K-P2x
C. De Concini and V. G. Kac, Representations of quantum groups at roots of 1, in Progr. Math., Vol. 92, pp. 471]506, Birkhauser, Basel, 1990. ¨ C. De Concini and V. G. Kac, Representations of quantum groups at roots of 1: Reduction to the exceptional case, Internat. J. Modern Phys. A 7 Ž1992., 141]149. C. De Concini, V. G. Kac, and C. Procesi, Quantum coadjoint action, J. Amer. Math. Soc. 5 Ž1992., 151]190. C. De Concini, V. G. Kac, and C. Procesi, Some remarkable degenerations of quantum groups, Comm. Math. Phys. 157 Ž1993., 405]427.
366 wD1x wD2x wF-P1x wF-P2x wHx wJx wK-W1x wK-W2x wL1x wL2x wPx wTx
N. CANTARINI
V. G. Drinfeld, Hopf algebras and quantum Yang]Baxter equation, So¨ iet Math. Dokl. 32 Ž1985., 254]258. V. G. Drinfeld, Quantum groups, in ‘‘Proc. ICM, Berkeley 1, 1986,’’ pp. 798]820. E. M. Friedlander and B. J. Parshall, Induction, deformation and specialization of Lie algebra representations, Math. Ann. 290 Ž1991., 473]489. E. M. Friedlander and B. J. Parshall, Modular representation theory of Lie algebras, Amer. J. Math. 110 Ž1988., 1055]1094. W. Hesselink, The normality of closures of orbits in a Lie algebra, Comment. Math. Hel¨ . 54 Ž1979., 105]110. M. Jimbo, A q-difference analogue of U Ž g . and the Yang]Baxter equation, Lett. Math. Phys. 10 Ž1985., 63]69. V. G. Kac and B. Weisfeiler, Irreducible representations of Lie p-algebras, Funct. Anal. Appl. 5 Ž1971., 28]36. V. G. Kac and B. Weisfeiler, Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic p, Indag. Math. 38 Ž1976., 135]151. G. Lusztig, Finite dimensional Hopf algebras arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 No. 1 Ž1990., 257]296. G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 Ž1990., 89]114. A. Premet, Irreducible representations of Lie algebras of reductive groups and the Kac]Weisfeiler conjecture, In¨ ent. Math. 121 Ž1995., 79]117. J. Tits, Sur les constants de structure et le theoreme d’existence des algebres de ´ Lie semi-simple, Inst. Hautes Etudes Sci. Publ. Math. 31 Ž1966., 21]58.
JOURNAL OF ALGEBRA ARTICLE NO.
Mod-p Reduction for Quantum Groups N. Cantarini Dipartimento di Matematica, Uni¨ ersita ` di Pisa, Via Buonarroti 2, Pisa, Italy Communicated by Corrado de Concini Received June 22, 1997
Let Ue Ž G . be the quantized enveloping algebra associated to the Lie algebra G s sl Ž n q 1. at a pth-root of unity « and assume that p is a prime which does not divide n q 1. It is known that the irreducible, finite dimensional representations of U« Ž G . are parametrized, up to isomorphisms, by the conjugacy classes of SLŽ n q 1.. In the paper we prove that the dimension of any U« Ž G .-module M parametrized by a conjugacy class O is divided by p1r 2 dimŽ O .. This result was conjectured by C. De Concini, V. G. Kac, and C. Procesi Ž J. Amer. Math. Soc. 5, 1992, 151]190.. Q 1998 Academic Press
INTRODUCTION Let Uq Ž G . be the quantized enveloping algebra associated to a simple, finite dimensional Lie algebra G according to the definition given by Drinfeld wD2x and Jimbo wJx, and let U« Ž G . be the algebra over C obtained from Uq Ž G . by specializing q at a primitive lth-root of unity « , l being a fixed odd integer strictly greater than 1. In the papers wDC-K1, DC-K-P1x it is shown how to parametrize the irreducible, finite dimensional representations of U« Ž G . by the conjugacy classes of the algebraic group G with Lie algebra G and trivial center. In particular it was conjectured that the dimension of any irreducible, finite dimensional U« Ž G .-module, corresponding to a conjugacy class O , is divisible by l 1r2 dimŽ O .. The main purpose of this paper is to prove this conjecture for the irreducible representations of the quantized enveloping algebra associated to G s sl Ž n q 1.. We recall that the regular case, that is, the case of the representations corresponding to a conjugacy class of maximal dimension, was proved by 357 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
358
N. CANTARINI
De Concini, Kac, and Procesi in wDC-K-P2x. According to wDC-K2x, any irreducible representation of U« Ž sl Ž n q 1.. is either unipotent Žsee Section 2 for the definition. or induced by a unipotent one. Therefore one can reduce to the study of the unipotent representations ŽSection 2.. The above conjecture is the quantum-analogue of a well known conjecture which was formulated by Kac and Weisfeiler in wK-W1x and proved by Premet in wPx. It states that if G is a simple and simply connected algebraic group defined over an algebraically closed field, whose characteristic p ) 0 is good for G Žsee Section 3. and such that G s LieŽ G . admits a non-degenerate G-invariant trace form, then any irreducible G-module V has dimension divisible by p1r2 dimŽ V Ž x .. , where x g G U is the p-character of V and V Ž x . is its coadjoint orbit in G U . We recall that, also for the Kac]Weisfeiler conjecture, a theorem of reduction to the nilpotent case, similar to the theorem mentioned above, had been proved in wK-W1x. We make use of Premet’s theorem to obtain our result. More precisely, we fix a prime p and a primitive pth-root of unity « and, starting from an irreducible U« Ž sl Ž n q 1..-module M parametrized by a unipotent conjugacy class O of SLŽ n q 1., we construct a representation M of the Lie algebra sl Ž n q 1. over an algebraically closed field of characteristic p, such that dimŽ M . s dimŽ M . and the dimension of O is equal to the dimension of the coadjoint orbit of the p-character of M. In order to realize this construction we will be considering the ‘‘limiting’’ specialization U1 of Uq Ž G . ŽSection 1.. Besides, taking into account Premet’s results wPx, we shall assume p ¦ n q 1. One fundamental step in our construction consists in proving that any irreducible, finite dimensional, unipotent U« Ž sl Ž n q 1..-module is a quotient of an induced module ŽSection 3.. I thank Professor Corrado De Concini for his precious suggestions and the long discussions we had.
1. DEFINITIONS AND NOTATIONS Let Ž a i j . i, js1, . . . , n be a symmetric Cartan matrix and let G be the associated simple, finite dimensional Lie algebra. By Q and Qq we shall denote the sets of roots and positive roots, respectively, with N s < Qq< , by R the root lattice, and by D a fixed set of simple roots. DEFINITION 1.1. The quantized enveloping algebra Uq Ž G . associated to Ž a i j . i,n js1 is the associative algebra over CŽ q . with generators Ei , Fi , K i" 1,
MOD-p REDUCTION FOR QUANTUM GROUPS
359
for i s 1, . . . , n, and the following defining relations: Ei Fj y Fj Ei s d i j K i K j s K j K i s K iqj , K i Ej Ky1 s q a i j Ej , i 1ya i j s Ý Ž y1.
ss0 1ya i j s Ý Ž y1.
ss0
K i y Ky1 i
Ž 1.1.
q y qy1
s Ky1 K i Ky1 i i Ki s 1
Ž 1.2.
K i Fj Ky1 s qya i j Fj i
Ž 1.3.
1 y a i j 1y a i jys Ei Ej Eis s 0 s
if i / j
Ž 1.4.
1 y a i j 1y a i jys Fi Fj Fi s s 0 s
if i / j.
Ž 1.5.
Here 1 ys a i j is the Gaussian binomial coefficient for example, wDC-K1x..
1 y ai j s d
with d s 1 Žsee,
Let W be the Weyl group associated to Ž a i j . and w 0 g W its element with maximal length: l Ž w 0 . s N. By TW we denote the corresponding Braid group whose generators Ti , with i s 1, . . . , n, act on Uq Ž G . according to wL2x. It is known that if we fix a reduced expression of w 0 : w 0 s si1 si 2 ??? si N , then we can define a total convex ordering in Qq by putting
b j s si1 ??? si jy 1Ž a i j . for j s 1, . . . , N. We define Eb j s Ti1 ??? Ti jy 1 Ei j ,
Fb j s Ti1 ??? Ti jy 1 Fi j
N and, for every k s Ž k 1 , . . . , k N . g Zq ,
E k s Ebk11 ??? EbkNN ,
F k s FbkNN ??? Fbk11 .
Let us fix a prime p and a primitive pth-root of unity « . By U« Ž G . we denote the associative algebra over C with generators Ei , Fi , K i" 1 and relations Ž1.1. ] Ž1.5. where q is replaced by « . Now put A s Zw q, qy1 x and take the A-subalgebra U A of Uq Ž G . generated by the elements Ei , Fi , K i , and Hi [ w Ei , Fi x, with the relations Ž1.1. ] Ž1.5. and K i y Ky1 s Ž q y qy1 . Hi . We define i U1 [ UAr Ž f qp , q y 1 . UA , where f qp is the pth cyclotomic polynomial in the variable q.
360
N. CANTARINI
Notice that the following isomorphism of fields holds: Z q, qy1 r Ž f qp , q y 1 . , ZrpZ. Furthermore we recall the following well known result: PROPOSITION 1.2 wL1x. U1 , UZr pZ Ž G . m ZrpZwŽZr2Z. r k G x where UZr pZ Ž G . is the uni¨ ersal en¨ eloping algebra of G o¨ er ZrpZ. 2. REDUCTION TO THE EXCEPTIONAL CASE Let RepŽ U« . be the set of the equivalence classes of the irreducible, finite dimensional representations of U« . Let Z« be the center of U« . Then Žsee wDC-K1x. the elements Eap , Fap , with a g Qq, and K ip , with i s 1, . . . , n, lie in Z« . Besides, if Z0 is the subalgebra of Z« generated by these elements, then U« is a free Z0-module with a basis consisting of the N vectors F k K 1m 1 ??? K nm n E r < k s Ž k 1 , . . . , k N ., r s Ž r 1 , . . . , rN . g Zq , mj g Z, 0 F k j , r j , m j F p y 14 . By G we shall denote the algebraic group whose Lie algebra is G and whose center is trivial. It is known wDC-K-P1x that the elements of RepŽ U« . are parametrized by the conjugacy classes of the group G as it is stated by the following theorem: THEOREM 2.1 wDC-K-P1x. There exists a map
w : Rep Ž U« . ª G such that: Ža. ImŽ w . is the big cell of G; Žb. the representations corresponding to conjugated elements in G are all of the same ‘‘type’’ Ž i.e., they are equi¨ alent, up to a twist, by the elements ˜ of automorphisms of U« ., in particular they ha¨ e the same of a group G dimension. In fact it has been shown by De Concini and Kac wDC-K2x that, in order to classify the irreducible, finite dimensional representations of U« , it is sufficient to consider the representations corresponding to the conjugacy classes of some special elements called exceptional. DEFINITION 2.2. A semisimple element g in G is called exceptional if the center of its centralizer ZG Ž g . in G is finite. DEFINITION 2.3. An element g in G is called exceptional if its semisimple part, with respect to the Jordan decomposition, is exceptional.
MOD-p REDUCTION FOR QUANTUM GROUPS
361
DEFINITION 2.4. A representation s in RepŽ U« . is called exceptional if w Ž s . is an exceptional element of G. Let us now take a non-exceptional representation s in RepŽ U« . on a vector space V, with central character x . Then we may regard s as a representation of the algebra Ux [ U«rŽ z y x Ž z ., z g Z0 .. As g s w Ž s . ˜ is not exceptional, we may suppose, up to the action of some element of G on s , that if g s g s g u is the Jordan decomposition of g, then g satisfies the following conditions: }g u belongs to the maximal unipotent subgroup Ny of G, corresponding to yQq; }h G [ LieŽcenterŽ ZG Ž g s ... / 0; }QX [ a g Q < a vanishes on h G 4 s ZDX l Q where ZDX is a sublattice of R spanned by a proper subset DX of D. Let U«X be the subalgebra of U« generated by the K i ’s, Ky1 i ’s, with i s 1, . . . , n and by the elements Ej , Fj such that a j g DX . Put UxX s U«Xr Ž z y x Ž z . , z g Z0 l U«X . , U g s U«X Uq,
Uxg s U gr Ž z y x Ž z . , z g Z0 l U g . ,
where Uq is the subalgebra of U« generated by the Ei ’s. Then the following theorem holds: THEOREM 2.5 wDC-K2x. There exists a unique irreducible Uxg-submodule V of V such that: X
Ža. V X is a UxX-module. Žb. The Ux-module V is induced by V X , V s Ux mUxg V X . In particular dimŽ V . s p t dimŽ V X ., where 2 t s < Q _ QX <. Žc. The map V ¬ V X is a bijection RepŽ U« . ª RepŽ UxX .. Remark. If we restrict the representation of UxX in V X to the subalgebra of UxX generated by the elements Ei , Fi , K i" 1 such that a i g DX , we get an irreducible representation. In fact this is an exceptional representation of the quantum group U« Ž G X . where G X is the subalgebra of G generated by its Chevalley generators corresponding to a i g DX . We recall that if G s sl Ž n q 1. the exceptional elements are the unipotent ones. Furthermore every unipotent matrix in SLŽ n. is conjugated to
362
N. CANTARINI
one with the form Jh 1 Ls
0
Jh 2
.. 0
. Jh t
0
,
Ž 2.6.
where h1 G h 2 G ??? G h t is a partition of n and Jt is a t = t Jordan block: 1 Jt s
^
1 .. .
0
..
. ..
0
.
`
1 1
0
.
_
t
We underline that this classification does not depend on the characteristic of the field.
3. THE A n-CASE In this section we shall examine the case G s sl Ž n q 1.. We know that the Weyl group W is the symmetric group Snq 1. From now on we shall assume that a reduced expression of w 0 , say w 0 s s n s ny 1 ??? s1 s n ??? s 2 s n ??? s n , and the associated ordering b 1 , . . . , bN 4 of the positive roots have been fixed. Let O be a conjugacy class of SLŽ n q 1. parametrized by a unipotent matrix L of type Ž2.6. for some blocks Jh1, . . . , Jh t , with h1 G h 2 G ??? G h t a partition of n q 1. Take an irreducible U« Ž G .-module M such that w Ž M . is conjugated to L. Our first aim is to show that every such a module is the quotient of an induced module. According to the definition of w Žsee wDC-K-P1x., for every positive root a we have Eap s 0, Fap either 0 or 1, and for every i s 1, . . . , n, K ip s 1 Žby Ea , Fa , K j we denote both the generators of sl« Ž n q 1. and their images through the representation .. We consider the set B s x g M < Ei Ž x . s 0 ; i s 1, . . . , n4 .
MOD-p REDUCTION FOR QUANTUM GROUPS
363
Since the subalgebra of sl« Ž n q 1. generated by the Ei ’s acts nilpotently, B is a non-trivial set upon which the K i ’s act diagonally; thus we can take a non-zero element m g B such that K i m s « s i m with 0 F si F p y 1. Now, if U U is the C-subalgebra of U« Ž sl Ž n q 1.. generated by Ei , K i , Fap for every i s 1, . . . , n, a g Qq, then the line L generated by m is an U Umodule. Since M is irreducible it is generated by m hence M is a quotient of the induced module I [ U« Ž sl Ž n q 1 . . mU U L. We have thus established the following proposition: PROPOSITION 3.1. Let M be an irreducible, finite dimensional U« Ž sl Ž n q 1..-module lying o¨ er a unipotent canonical element of SLŽ n q 1.. Then there exist a proper subalgebra U U of U« and an element m g M _ 04 such that, if L is the line generated by m, then: Ž1. Ž2. of I.
L is an U U -module; M s IrS where I [ U« Ž sl Ž n q 1.. mU U L and S is a submodule
Remark. The induced module I has got a natural basis C : C s Fbh11 ??? FbhNN m < 0 F h j - p 4 . If C s Zw « x-span C 4 then C is stable with respect to the action of the Ei ’s, Fi ’s, K i ’s, Hi ’s for every i s 1, . . . , n. Indeed, for every z in C such that z K j z s « l j z Ž0 F l jz F p y 1. the following relations hold: Ej Fjk z s Fjk Ej z q Ž1 y« 2 k . Ž « 2y2 kq l j y «y l j .rŽ1 y« 2 .Ž « y«y1 . z
v
Fjky 1 z;
z
1 k Ej Fhk. . . j z s yŽ1 y « 2 k .rŽ1 y « 2 . «y l j Fhky . . . j Fh . . . jy1 z q Fh . . . j Ej z if z
v
h - j;
k Ej Fjk. . . t z s Ž1 y «y2 k .rŽ1 y «y2 . « l j q1 Fjky1 . . . t Fjq1 . . . t z q Fj . . . t E j z if z
v
t ) j; v v v v v
Ej Fhk. . . t s Fhk. . . t Ej if h - j - t; Fj Fhk. . . t s Fhk. . . t Fj if h - j - t; Fj Fjk. . . t s « k Fjk. . . t Fj if t ) j; Fj Fhk. . . j s «yk Fhk. . . j Fj if h - j; k yk k ky1 Fj Fjq1 Fjq1 . . . t Fj q «yk Ž1 y « 2 k .rŽ1 y « 2 . Fjq1 ... t s « . . . t Fj . . . t .
We notice that, since U« Ž sl Ž n q 1.. is a finitely generated algebra, the module M is defined over a finite extension K of QŽ « . and, in order to compute the dimension of M, we may reduce to K, instead of C.
364
N. CANTARINI
Let L be the localization of the ring of integers Zw « x of QŽ « . at the ideal P0 generated by « y 1, by m P 0 denote the maximal ideal of L , and let U L s UArf qp UA mZw « x L . Then the induced module I is defined over U L . Let us choose an ideal P, in the ring of integers Z of K, lying over P0 and let L X be the localization of Z at P, m P its maximal ideal. Then L Xrm P is a finite extension of Lrm P 0 and thus a finite field of characteristic p. We put U L X s U L mL L X . PROPOSITION 3.2. Let M be an irreducible U« Ž sl Ž n q 1..-module such that w Ž M . is a unipotent canonical element in SLŽ n q 1.. Then there exist a finite extension K of QŽ « ., a local ring L X in K, and an U L X-module M X such that M X is a free L X-module and M s M X mL X K . Proof. We use the same notations as in Proposition 3.1; then we know that M is the quotient of the induced module I by a submodule S. According to the remark on the basis C of I, we have I s I P mL X K , where I P is the module, over the localization L X of Z in P, naturally defined. Now, since L X is a discrete valuation ring, S l I P is free and it is a direct factor of I P , hence, if we put M X s I P rŽ S l I P . we obtain the result. In particular, rkŽ S l I P . s dim K S and dim K M s rkŽ M X .. Notice that the action of U« on M is obtained by extending that of UL X . Before coming to a conclusion we recall the following theorem: THEOREM 3.3 wPx. Let G be a simple and simply connected algebraic group defined o¨ er an algebraically closed field whose characteristic p ) 0 is good for G. Suppose that G s LieŽ G . admits a non-degenerate G-in¨ ariant trace form. Let V be an irreducible G-module with p-character x g G U . Then the dimension of V is di¨ isible by p1r2 dimŽ V Ž x .. , where V Ž x . is the coadjoint orbit of x . Here by a ‘‘good’’ prime we mean a prime greater than any coefficient of any positive root in its expression relative to a basis of simple roots. Therefore if G s sl Ž n q 1. any prime p is good for G . On the other hand, in order to guarantee the existence of a non-degenerate trace form on G , we shall suppose p ¦ n q 1. In fact in wPx it is shown that the
MOD-p REDUCTION FOR QUANTUM GROUPS
365
Kac]Weisfeiler conjecture is no longer true for G s sl Ž mp, k. when m G 1 Ž p is the characteristic of the field k.. We are now ready to state our main result: THEOREM 3.4. Let « be a primiti¨ e pth-root of unity with p prime, p ) 2, p ¦ n q 1. If M is an irreducible, finite dimensional U« Ž sl Ž n q 1..-module lying o¨ er a conjugacy class O , then dimŽ M . s kp1r2 dimŽ O . for some positi¨ e integer k. Proof. As we explained in Section 2 we may reduce to the unipotent case and thus suppose that O is the conjugacy class of a unipotent element of SLŽ n q 1.. Then let us assume the construction of the module M X and, ˜ s M Xrm P M X . As L X is a using the same notations as above, let us put M X ˜ .. Besides, by its own definition, M˜ is a local ring, rkŽ M . s dim L X r m PŽ M U1-module and therefore a UZr pZ Ž G .-module Žsee Section 1.. We finally denote the algebraic closure of the finite field L Xrm P by L Xrm P and define
˜ mL X r m P L Xrm P . MsM If x is the p-character of M, then dim K Ž M . s dim L Xrm P Ž M . s kp1r2 dimŽ V Ž x .. for some positive integer k. Indeed, since M is finite-dimensional, it has a composition series consisting of irreducible modules M j satisfying the hypotheses of Theorem 3.3, whose dimension is therefore divided by p1r2 dimŽ V Ž x ... We point out that the canonical unipotent element L of type Ž2.6., corresponding to the Jordan partition l, reduces ‘‘modulo p’’ to the canonical nilpotent element corresponding to l, that is, to the nilpotent element with the same blocks as L but with all zeros on the diagonal. Therefore the dimension of the coadjoint orbit V Ž x . of x is the dimension of the conjugacy class O .
REFERENCES wDC-K1x wDC-K2x wDC-K-P1x wDC-K-P2x
C. De Concini and V. G. Kac, Representations of quantum groups at roots of 1, in Progr. Math., Vol. 92, pp. 471]506, Birkhauser, Basel, 1990. ¨ C. De Concini and V. G. Kac, Representations of quantum groups at roots of 1: Reduction to the exceptional case, Internat. J. Modern Phys. A 7 Ž1992., 141]149. C. De Concini, V. G. Kac, and C. Procesi, Quantum coadjoint action, J. Amer. Math. Soc. 5 Ž1992., 151]190. C. De Concini, V. G. Kac, and C. Procesi, Some remarkable degenerations of quantum groups, Comm. Math. Phys. 157 Ž1993., 405]427.
366 wD1x wD2x wF-P1x wF-P2x wHx wJx wK-W1x wK-W2x wL1x wL2x wPx wTx
N. CANTARINI
V. G. Drinfeld, Hopf algebras and quantum Yang]Baxter equation, So¨ iet Math. Dokl. 32 Ž1985., 254]258. V. G. Drinfeld, Quantum groups, in ‘‘Proc. ICM, Berkeley 1, 1986,’’ pp. 798]820. E. M. Friedlander and B. J. Parshall, Induction, deformation and specialization of Lie algebra representations, Math. Ann. 290 Ž1991., 473]489. E. M. Friedlander and B. J. Parshall, Modular representation theory of Lie algebras, Amer. J. Math. 110 Ž1988., 1055]1094. W. Hesselink, The normality of closures of orbits in a Lie algebra, Comment. Math. Hel¨ . 54 Ž1979., 105]110. M. Jimbo, A q-difference analogue of U Ž g . and the Yang]Baxter equation, Lett. Math. Phys. 10 Ž1985., 63]69. V. G. Kac and B. Weisfeiler, Irreducible representations of Lie p-algebras, Funct. Anal. Appl. 5 Ž1971., 28]36. V. G. Kac and B. Weisfeiler, Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic p, Indag. Math. 38 Ž1976., 135]151. G. Lusztig, Finite dimensional Hopf algebras arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 No. 1 Ž1990., 257]296. G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 Ž1990., 89]114. A. Premet, Irreducible representations of Lie algebras of reductive groups and the Kac]Weisfeiler conjecture, In¨ ent. Math. 121 Ž1995., 79]117. J. Tits, Sur les constants de structure et le theoreme d’existence des algebres de ´ Lie semi-simple, Inst. Hautes Etudes Sci. Publ. Math. 31 Ž1966., 21]58.