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Character values, Schur indices and character sheaves


Author: Meinolf Geck
Journal: Represent. Theory 7 (2003), 19-55
MSC (2000): Primary 20C15; Secondary 20G40
DOI: https://doi.org/10.1090/S1088-4165-03-00170-5
Published electronically: February 19, 2003
MathSciNet review: 1973366
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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 $q$) these values lie in the ring of algebraic integers generated by $(1+ \sqrt{\pm q})/2$ and roots of unity of order prime to $q$. 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.


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  • 1. W. M. BEYNON AND N. SPALTENSTEIN, Green functions of finite Chevalley groups of type $E_n$ $(n=6,7,8)$, J. of Algebra 88 (1984), 584-614. MR 85k:20136
  • 2. C. BONNAFÉ, Mackey formula in type $A$, 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 $2$ 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 $(F_4)$, 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 $F_4$ in guter Charakteristik, Ph.D. Thesis, RWTH Aachen, 1995.

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Additional Information

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: https://doi.org/10.1090/S1088-4165-03-00170-5
Received by editor(s): July 15, 2002
Received by editor(s) in revised form: November 13, 2002
Published electronically: February 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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