Representation Theory

Author(s): Gunter Malle
Journal: Represent. Theory 11 (2007), 192-220.
MSC (2000): Primary 20C33, 20G40
Posted: December 5, 2007
Retrieve article in: PDF DVI PostScript

Abstract: We give a classification of irreducible characters of finite groups of Lie type of

-degree, where

is any prime different from the defining characteristic, in terms of local data. More precisely, we give a classification in terms of data related to the normalizer of a suitable Levi subgroup, which in many cases coincides with the normalizer of a Sylow

-subgroup. The McKay conjecture asserts that there exists a bijection between characters of

-degree of a group and of the normalizer of a Sylow

-subgroup. We hope that our result will constitute a major step towards a proof of this conjecture for groups of Lie type, and, in conjunction with a recent reduction result of Isaacs, Malle and Navarro, for arbitrary finite groups.

1.: M. BROUÉ AND G. MALLE, Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis. Math. Ann. 292 (1992), 241-262. MR 1149033 (93a:20076)
2.: M. BROUÉ AND G. MALLE, Zyklotomische Heckealgebren. Astérisque 212 (1993), 119-189. MR 1235834 (94m:20095)
3.: M. BROUÉ AND G. MALLE, Generalized Harish-Chandra theory, in: Representations of reductive groups, Cambridge University Press, Cambridge, 1998, pp. 85-103. MR 1714151 (2000k:20063)
4.: M. BROUÉ, G. MALLE AND J. MICHEL, Generic blocks of finite reductive groups. Astérisque 212 (1993), 7-92. MR 1235832 (95d:20072)
5.: M. BROUÉ, G. MALLE AND J. MICHEL, Towards spetses. I. Transform. Groups 4 (1999), 157-218. MR 1712862 (2001b:20082)
6.: M. BROUÉ AND J. MICHEL, Blocs à groupes de défaut abéliens des groupes réductifs finis. Astérisque 212 (1993), 93-117. MR 1235833 (94j:20007)
7.: M. CABANES, Unicité du sous-groupe abélien distingué maximal dans certains sous-groupes de Sylow. C. R. Acad. Sci. 318 (1994), 889-894. MR 1278146 (95b:20035)
8.: R.W. CARTER, Finite groups of Lie type. Wiley-Interscience, New York, 1985. MR 794307 (87d:20060)
9.: F. DIGNE AND J. MICHEL, On Lusztig's parametrization of characters of finite groups of Lie type. Astérisque 181-182 (1990), 113-156. MR 1051245 (91d:20041)
10.: F. DIGNE AND J. MICHEL, Representations of finite groups of Lie type. London Math. Soc. Student Texts, Cambridge University Press, Cambridge, 1991. MR 1118841 (92g:20063)
11.: F. DIGNE AND J. MICHEL, Groupes réductifs non connexes. Ann. Sci. École Norm. Sup. 27 (1994), 345-406. MR 1272294 (95f:20068)
12.: M. GECK AND G. HISS, Basic sets of Brauer characters of finite groups of Lie type. J. Reine Angew. Math. 418 (1991), 173-188. MR 1111205 (92e:20006)
13.: D. GORENSTEIN AND R. LYONS, The local structure of finite groups of characteristic type. Mem. Amer. Math. Soc. 42 (1983), no. 276. MR 690900 (84g:20025)
14.: D. GORENSTEIN, R. LYONS AND R. SOLOMON, The classification of the finite simple groups, Number 3. Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, 1994. MR 1303592 (95m:20014)
15.: M. ISAACS, G. MALLE, G. NAVARRO, A reduction theorem for the McKay conjecture. Invent. Math. 170 (2007), 33-101.
16.: P. KLEIDMAN, The maximal subgroups of the Steinberg triality groups ${}^3D_ 4(q)$ and of their automorphism groups. J. Algebra 115 (1988), 182-199. MR 937609 (89f:20024)
17.: G. LUSZTIG, On the representations of reductive groups with disconnected centre, in: Orbites unipotentes et représentations, I. Astérisque 168, (1988), pp. 157-166. MR 1021495 (90j:20083)
18.: G. MALLE, Die unipotenten Charaktere von ${}^2F_4(q^2)$ . Comm. Algebra 18 (1990), 2361-2381. MR 1063140 (91k:20015)
19.: G. MALLE, Darstellungstheoretische Methoden bei der Realisierung einfacher Gruppen vom Lie Typ als Galoisgruppen, in: Representation theory of finite groups and finite-dimensional algebras, Progr. Math., 95, Birkhäuser, Basel, 1991, pp. 443-459. MR 1112174 (93a:12005)
20.: G. MALLE, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen. J. Algebra 177 (1995), 768-826. MR 1358486 (97a:20073)
21.: G. MALLE, Degrés relatifs des algèbres cyclotomiques associées aux groupes de réflexions complexes de dimension deux, in: M. Cabanes (ed.), Finite reductive groups, related structures and representations, Progress in Math. 141, Birkhäuser, 1997, pp. 311-332. MR 1429878 (98h:20019)
22.: G. MALLE, On the generic degrees of cyclotomic algebras. Represent. Theory 4 (2000), 342-369. MR 1773866 (2002a:20005)
23.: G. MALLE, The inductive McKay condition for simple groups not of Lie type, to appear in Comm. Algebra.
24.: B. SPÄTH, Die McKay-Vermutung für quasi-einfache Gruppen vom Lie-Typ. Dissertation, TU Kaiserslautern, 2007.
25.: B. SPÄTH, The McKay-conjecture for exceptional groups and odd primes, submitted, 2007.
26.: T. SPRINGER AND R. STEINBERG, Conjugacy classes, in: A. Borel et al. (eds.), Seminar on algebraic groups and related finite groups, Lecture Notes in Math. 131, Springer, Berlin Heidelberg New York, 1970, pp. 167-266. MR 0268192 (42:3091)
27.: R. STEINBERG, Lectures on Chevalley groups. Mimeographed notes, Yale University, New Haven, CT, 1968. MR 0466335 (57:6215)
28.: R. STEINBERG, Endomorphisms of linear algebraic groups. Memoirs of the Amer. Math. Soc. 80, Amer. Math. Soc., Providence, RI, 1968. MR 0230728 (37:6288)
29.: R. STEINBERG, Torsion in reductive groups. Advances in Math. 15 (1975), 63-92. MR 0354892 (50:7369)
30.: L.C. WASHINGTON, Introduction to cyclotomic fields. 2nd ed. Graduate Texts in Math. 83, Springer, New York, 1997. MR 1421575 (97h:11130)

Gunter Malle
Affiliation: FB Mathematik, Universität Kaiserslautern, Postfach 3049, D 67653 Kaiserslautern, Germany
Email: malle@mathematik.uni-kl.de

DOI: 10.1090/S1088-4165-07-00312-3
PII: S 1088-4165(07)00312-3
Received by editor(s): April 7, 2006
Received by editor(s) in revised form: September 16, 2007
Posted: December 5, 2007
Dedicated: Dedicated to Professor Toshiaki Shoji on the occasion of his 60th birthday
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-4165

Previous volume \| Table of contents \| Next volume Articles in press \| Most recent volume \| All volumes Previous Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS