×

zbMATH — the first resource for mathematics

Representations of generic algebras and finite groups of Lie type. (English) Zbl 0537.20018
Let G be the group of \({\mathbb{F}}_ q\)-rational points of a connected reductive \({\mathbb{F}}_ q\)-group G, where \({\mathbb{F}}_ q\) is the Galois field of q elements. The group G has a split (B,N)-pair in characteristic p \((q=p^ e)\). Each standard parabolic subgroup \(P_ J\) of G (where J is a subset of the simple roots associated with the (B,N)-structure) has a standard Levi decomposition \(P_ J=M_ JU_ J\), where \(U_ J\) is the ”unipotent radical” of \(P_ J\), and \(M_ J\) the standard Levi component of \(P_ J.\)
From the introduction: ”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 W. More precisely, the ”Harish-Chandra” principle (§ 3) organizes the representations of G into ”series” which consist of the irreducible constituents of a fixed induced cuspidal representation \(Ind(P_ L\to G;\tilde D)\) (where \(P_ L\) is a parabolic subgroup of G and D is a cuspidal representation of \(M_ L)\). In [Invent. Math. 58, 37-64 (1980; Zbl 0435.20023)], the constituents of a single series where studied by elucidating the structure of the endomorphism algebra of \(Ind(P_ L\to G;\tilde D).\) In § 1 of the present work we make explicit the connection between the decomposition of induced representations and the representation theory of the associated endomorphism algebras by introducing two pairs of adjoint functors and studying the way they interact. This transfers the study of any particular series to the study of a certain endomorphism algebra.”
The principal result of the paper cited above was that the endomorphism algebras concerned have an explicit presentation in terms of the ramification group W(D) of D and certain powers \(p_ a\) of the characteristic. ”Replacing these \(p_ a\) by indeterminates \(u_ a\), one constructs ”generic algebras” over the ring \({\mathbb{C}}[\{u_ a\}]\); § 4 of the present work is devoted to the study of these generic algebras and of the connection between their representation theory and that of the ramification groups W(D). This provides (Theorem (4.8)) explicit descriptions of each of the series of representations of G, in terms of data relating to the cuspidal representation and its ramification group. In § 5 a refinement (5.6) of this description is proved, in which G is replaced by the Levi component \(M_ J\) of the parabolic subgroup \(P_ J.''\)
”In general terms, the Comparison Theorem (5.9) states that if \(\phi\) and \(\psi\) are characters of appropriate projective representations of the ramification group \(W(\delta)\) and its subgroup \(W^ w_ J\cap W(\delta)\) (where \(wL\subset J\subset \Pi\) and \(\delta\) is an irreducible cuspidal character of \(M_ L)\), and \(\zeta_{\phi}\), \(\eta_{w\psi}\) are the corresponding characters in the \(\delta\)-series of G and \(w\delta\)-series of \(M_ J\) respectively, then the multiplicities of \(\zeta_{\phi}\) in \(Ind(P_ J\to G;{\tilde \eta}_{w\psi})\) and of \(\phi\) in \(Ind(W^ w_ J\cap W(\delta)\to W(\delta);\psi)\) are equal. The proof is accomplished by showing that both multiplicities are equal to a certain multiplicity which arises in the generic algebra context.”
As an application of this ”comparison theorem” the authors are able to determine precisely what the dual of an arbitrary irreducible character of G is, thus generalizing the results of C. W. Curtis [J. Algebra 62, 320-332 (1980; Zbl 0426.20006)].
Reviewer: G.Noskov

MSC:
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C30 Representations of finite symmetric groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Dean Alvis, The duality operation in the character ring of a finite Chevalley group, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 907 – 911. · Zbl 0485.20029
[2] N. Bourbaki, Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles, No. 1308, Hermann, Paris, 1964 (French). · Zbl 0547.13002
[3] Charles W. Curtis, Truncation and duality in the character ring of a finite group of Lie type, J. Algebra 62 (1980), no. 2, 320 – 332. · Zbl 0426.20006 · doi:10.1016/0021-8693(80)90185-4 · doi.org
[4] Charles W. Curtis, Reduction theorems for characters of finite groups of Lie type, J. Math. Soc. Japan 27 (1975), no. 4, 666 – 688. · Zbl 0382.20007 · doi:10.2969/jmsj/02740666 · doi.org
[5] C. W. Curtis, N. Iwahori, and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with (\?, \?)-pairs, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 81 – 116. · Zbl 0254.20004
[6] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. · Zbl 0131.25601
[7] R. B. Howlett and G. I. Lehrer, A comparison theorem and other formulae in the character ring of a finite group of Lie type, Papers in algebra, analysis and statistics (Hobart, 1981) Contemp. Math., vol. 9, Amer. Math. Soc., Providence, R.I., 1981, pp. 285 – 288. · Zbl 0488.20033
[8] R. B. Howlett and G. I. Lehrer, Duality in the normalizer of a parabolic subgroup of a finite Coxeter group, Bull. London Math. Soc. 14 (1982), no. 2, 133 – 136. · Zbl 0482.20029 · doi:10.1112/blms/14.2.133 · doi.org
[9] R. B. Howlett and G. I. Lehrer, Induced cuspidal representations and generalised Hecke rings, Invent. Math. 58 (1980), no. 1, 37 – 64. · Zbl 0435.20023 · doi:10.1007/BF01402273 · doi.org
[10] G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), no. 2, 125 – 175. · Zbl 0372.20033 · doi:10.1007/BF01390002 · doi.org
[11] Nagayoshi Iwahori, Generalized Tits system (Bruhat decompostition) on \?-adic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 71 – 83.
[12] Kathy McGovern, Multiplicities of principal series representations of finite groups with split (\?,\?)-pairs, J. Algebra 77 (1982), no. 2, 419 – 442. · Zbl 0492.20023 · doi:10.1016/0021-8693(82)90264-2 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.