Representations of generic algebras and finite groups of Lie type
Representations of generic algebras and finite groups of Lie type
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DOI：
10.1090/s0002-9947-1983-0716849-6
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发表时间：
1983-02
影响因子：
1.3
通讯作者：
R. Howlett;G. Lehrer
中科院分区：
文献类型：
--
作者：
R. Howlett;G. Lehrer
有限李群G的复表示论与某些“泛代数”的复表示论有关。因此，导出了公式（“比较定理”），将 G 中的重数与 G 的 Weyl 群 W 中的重数联系起来。应用包括 G 的任意不可约复表示的对偶（见下文）的显式描述。 简介。令 G 为（连通的）还原 F(/-群 G 的 F^-有理点群，其中 F^ 是 q 元素的伽罗瓦域。本文的目的是使用我们之前的论文 [9] 中暗示的“通用代数”技术来研究 G 的复表示理论，目前的工作可被视为续集。对于任何未解释的符号，读者可以参考 [9]。我们的主要定理 (5.9) 比较了内G 的某些特征与其 Weyl 群 W 内的相应内积的乘积。作为此“比较定理”的应用，我们精确地确定 G 的任意不可约特征的对偶（参见 Curtis [3]）。 J 是与 (/?, N )-结构相关的简单根的子集，具有标准 Levi 分解 Pj = MjUj，其中 Uj 是 Pj 的“单能根”，而 Mj（Pj 的标准 Levi 分量）又是还原性 F^-群的有理点群，读者可以参考 [9, §1]。 “泛代数”，是复数多项式环上的结合代数，“专门”于对 G 和 IF 的表示论具有重要意义的各种复代数。更准确地说，“Harish-Chandra 原理”（参见下面的 [9，§1] 或 §3）将 G 的表示组织成“级数”，其中由固定诱导尖头表示 Ind(P7 -» 的不可约成分组成。 G; D)（其中 PL 是 G 的抛物线子群，D 是 ML 的尖点表示）。在[9]中，通过阐明 IndiP 的自同态代数的结构来研究单个级数的组成部分，-» D)。在本工作的第 1 节中，我们明确了归纳表征的分解与编辑 1982 年 12 月 20 日收到的表征理论之间的联系。1980 年数学学科分类。主要20GO5、20G40；次级 16A64、16A65。 ©1983 美国数学会 0002-9947/83 $1.00 + $.25 每页
The complex representation theory of a finite Lie group G is related to that of certain "generic algebras". As a consequence, formulae are derived ("the Comparison Theorem"), relating multiplicities in G to multiplicities in the Weyl group W of G. Applications include an explicit description of the dual (see below) of an arbitrary irreducible complex representation of G. Introduction. Let G be the group of F^-rational points of a (connected) reductive F(/-group G, where F^ is the Galois field of q elements. The purpose of this paper is to study the complex representation theory of G using the "generic algebra" techniques hinted at in our earlier paper [9], to which the present work may be considered a sequel. For any unexplained notation, the reader is referred to [9]. Our main theorem (5.9) compares inner products of certain characters of G with corresponding inner products within its Weyl group W. As an application of this "comparison theorem" we determine precisely what the dual (cf. Curtis [3]) of an arbitrary irreducible character of G is. In order to describe our results explicitly, we mention the principal facts concerning the structure of G which we shall require. Firstly, G has a split (/?, N)-oair in characteristic p (where q = pe). Secondly, each standard parabolic subgroup Pj of G (where J is a subset of the simple roots associated with the (/?, N )-structure) has a standard Levi decomposition Pj = MjUj, where Uj is the "unipotent radical" of Pj, and Mj, the standard Levi component of Pj, is again the group of rational points of a reductive F^-group. For more details, the reader is referred to [9, §1]. The passage between the representation theory of G and of its Weyl group W is accomplished by means of certain "generic algebras", which are associative algebras over a complex polynomial ring and which "specialize" to various complex algebras which have significance for the representation theory of G and IF. More precisely, the "Harish-Chandra principle" (see [9, §1] or §3 below) organizes the representations of G into "series" which consist of the irreducible constituents of a fixed induced cuspidal representation Ind(P7 -» G; D) (where PL is a parabolic subgroup of G and D is a cuspidal representation of ML). In [9], the constituents of a single series were studied by elucidating the structure of the endomorphism algebra of IndiP, -» G; D). In §1 of the present work we make explicit the connection between the decomposition of induced representations and the representation theory of the Received by the editors December 20, 1982. 1980 Mathematics Subject Classificution. Primary 20GO5, 20G40; Secondary 16A64, 16A65. ©1983 American Mathematical Society 0002-9947/83 $1.00 + $.25 per page