Representation Theory

Author(s): Meinolf Geck
Journal: Represent. Theory 7 (2003), 19-55.
MSC (2000): Primary 20C15; Secondary 20G40
Posted: February 19, 2003
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Abstract: In this paper we are concerned with the problem of determining the character values and Schur indices of a finite group of Lie type over ${\mathbb F}_q$ . We show that (under some conditions on

) these values lie in the ring of algebraic integers generated by $(1+ \sqrt{\pm q})/2$ and roots of unity of order prime to

. Furthermore, we determine the Schur indices for some of the (nonrational) unipotent characters in exceptional groups. Our results, combined with previous results due to Gow, Ohmori and Lusztig, imply that there are only 6 cases left where the Schur index of a cuspidal unipotent character remains unknown. Our methods rely, in an essential way, on Lusztig's theory of character sheaves.

1.: W. M. BEYNON AND N. SPALTENSTEIN, Green functions of finite Chevalley groups of type , J. of Algebra 88 (1984), 584-614. MR 85k:20136
2.: C. BONNAFÉ, Mackey formula in type , Proc. London Math. Soc. 80 (2000), 545-574. MR 2001i:20100
3.: R. W. CARTER, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, New York (1985). MR 94k:20020
4.: J. H. CONWAY, R. T. CURTIS, S. P. NORTON, R. A. PARKER AND R. A. WILSON, Atlas of finite groups, Clarendon Press 1985, Oxford. MR 88f:20002
5.: C. W. CURTIS AND I. REINER, Methods of representation theory, vol. II, Wiley, New York, 1987; reprinted 1994 as Wiley Classics Library Edition. MR 98i:20015
6.: F. DIGNE, G. I. LEHRER AND J. MICHEL, On Gel'fand-Graev characters of reductive groups with disconnected centre, J. Reine Angew. Math. 491 (1997), 131-147. MR 87h:20071
7.: F. DIGNE AND J. MICHEL, Fonctions ${\mathcal{L}}$ des variétés de Deligne-Lusztig et descente de Shintani, Mém. Soc. Math. France (N.S.), no. 20, 113, 1985. MR 92g:20063
8.: F. DIGNE AND J. MICHEL, Representations of finite groups of Lie type, London Math. Soc. Students Texts, 21, Cambridge University Press, 1991. MR 83k:20013
9.: P. FONG AND B. SRINIVASAN, The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982), 109-153. MR 96a:20020
10.: M. GECK, Basic sets of Brauer characters of finite groups of Lie type, III. Manuscripta Math. 85 (1994), 195-216. MR 98c:20084
11.: -, On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes, Doc. Math. J. DMV 1 (1996), 293-317 (electronic). MR 99m:20017
12.: M. GECK, G. HISS, F. L¨UBECK, G. MALLE AND G. PFEIFFER, CHEVIE-A system for computing and processing generic character tables, in: Computational methods in Lie theory, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175-210. MR 2002k:20017
13.: M. GECK AND G. PFEIFFER, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monographs, New Series 21, Oxford University Press, 2000. MR 57:6210
14.: R. GOW, Schur indices of some groups of Lie type, J. Algebra 42 (1976), 102-120. MR 57:417
15.: I. M. ISAACS, Character theory of finite groups, Academic Press, 1976; reprinted 1994 by Dover Publications. MR 96j:11137
16.: G. J. JANUSZ, Algebraic number fields. Second edition, Graduate Studies in Math., vol. 7, Amer. Math. Soc., Providence, RI, 1996. MR 88a:20058
17.: N. KAWANAKA, Generalized Gelfand-Graev representations of exceptional algebraic groups I, Invent. Math. 84 (1986), 575-616. MR 56:12138
18.: G. LUSZTIG, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976), 101-159. MR 57:3228
19.: -, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), 125-175. MR 57:3228
20.: -, Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics, vol. 39, Amer. Math. Soc., Providence, RI, 1977. MR 80f:20045
21.: -, Characters of reductive groups over a finite field, Annals Math. Studies 107, Princeton University Press (1984). MR 86j:20038
22.: -, Intersection cohomology complexes on a reductive group, Invent. Math. 75, 205-272 (1984). MR 86d:20050
23.: -, Character sheaves, Advances in Math. 56, 193-237 (1985); II, 57, 226-265 (1985); III, 57, 266-315 (1985); IV, 59, 1-63 (1986); V, 61, 103-155 (1986). MR 87b:20055; MR 87m:20118a; MR 87m:20118b; MR 87m:20118c
24.: -, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 157-166. MR 90j:20083
25.: -, Green functions and character sheaves, Annals of Math. 131 (1990), 355-408. MR 91c:20054
26.: -, A unipotent support for irreducible representations, Advances in Math. 94, 139-179 (1992). MR 94a:20073
27.: -, Remarks on computing irreducible characters, J. Amer. Math. Soc. 5, 971-986 (1992). MR 93g:20092
28.: -, Rationality properties of unipotent representations, J. Algebra 258 (2002), 1-22.
29.: G. MALLE, Die unipotenten Charaktere von ${^2F}_4(q^2)$ , Comm. Algebra 18 (1990), 2361-2381. MR 91k:20015
30.: Z. OHMORI, On the Schur indices of certain irreducible characters of reductive groups over finite fields, Osaka J. Math 25 (1988), 149-159. MR 89d:20034
31.: -, On the existence of characters of the Schur index of the simple finite Steinberg groups of type ${^2E}_6$ , Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 296-298. MR 94i:20079
32.: -, The Schur indices of the cuspidal unipotent characters of the finite unitary groups, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 111-113. MR 97i:20018
33.: K. SHINODA, The conjugacy classes of the finite Ree groups of type , J. Fac. Sci. Univ. Tokyo 22 (1975), 1-15. MR 51:8281
34.: T. SHOJI, Character sheaves and almost characters of reductive groups, I, Advances in Math. 111 (1995), 244-313. MR 95k:20069
35.: -, Character sheaves and almost characters of reductive groups, II, Advances in Math. 111 (1995), 314-354. MR 95k:20069
36.: -, Unipotent characters of finite classical groups, in: Finite reductive groups: Related structures and representations (ed. M. Cabanes), pp. 373-413; Birkhäuser Boston, 1997. MR 98h:20079
37.: N. SPALTENSTEIN, On the generalized Springer correspondence for exceptional groups, Advanced Studies in Pure Math. 6, North-Holland, Amsterdam (1985), 317-338. MR 87g:20072b
38.: T. A. SPRINGER AND R. STEINBERG, Conjugacy classes, in: A. BOREL ET AL., Seminar on Algebraic Groups and Related Finite Groups, Springer Lecture Notes in Mathematics Vol. 131, Springer Verlag, Berlin Heidelberg, 1970. MR 42:3091
39.: M. SUZUKI, On a class of doubly transitive groups, Annals of Math. 75 (1962), 105-145. MR 25:112
40.: P. H. TIEP AND A. E. ZALESSKI, Strong rationality of unipotent elements and realization fields of complex representations of finite groups of Lie type, J. Algebra, to appear.
41.: A. TURULL, The Schur indices of the irreducible characters of the special linear groups, J. Algebra 235 (2001), 275-314. MR 2001m:20011
42.: E. WINGS, Über die unipotenten Charaktere der Chevalley-Gruppen vom Typ in guter Charakteristik, Ph.D. Thesis, RWTH Aachen, 1995.

Meinolf Geck
Affiliation: Institut Girard Desargues, bat. Jean Braconnier, Université Lyon 1, 21 av Claude Bernard, F--69622 Villeurbanne Cedex, France
Email: geck@desargues.univ-lyon1.fr

DOI: 10.1090/S1088-4165-03-00170-5
PII: S 1088-4165(03)00170-5
Received by editor(s): July 15, 2002 and, in revised form, November 13, 2002
Posted: February 19, 2003
Copyright of article: Copyright 2003, American Mathematical Society

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