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Character values, Schur indices and character sheaves. (English) Zbl 1065.20019
Summary: 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.

MSC:
20C33 Representations of finite groups of Lie type
20C15 Ordinary representations and characters
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
Software:
CHEVIE
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References:
[1] W. M. Beynon and N. Spaltenstein, Green functions of finite Chevalley groups of type \?_{\?} (\?=6,7,8), J. Algebra 88 (1984), no. 2, 584 – 614. · Zbl 0539.20025
[2] Cédric Bonnafé, Mackey formula in type A, Proc. London Math. Soc. (3) 80 (2000), no. 3, 545 – 574. · Zbl 1037.20051
[3] Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication.
[4] Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. · Zbl 0616.20001
[5] F. Digne, G. I. Lehrer, and J. Michel, On Gel\(^{\prime}\)fand-Graev characters of reductive groups with disconnected centre, J. Reine Angew. Math. 491 (1997), 131 – 147. · Zbl 0880.20033
[6] François Digne and Jean Michel, Fonctions \? des variétés de Deligne-Lusztig et descente de Shintani, Mém. Soc. Math. France (N.S.) 20 (1985), iv+144 (French, with English summary). · Zbl 0608.20027
[7] François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. · Zbl 0815.20014
[8] Paul Fong and Bhama Srinivasan, The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982), no. 1, 109 – 153. · Zbl 0507.20007
[9] Meinolf Geck, Basic sets of Brauer characters of finite groups of Lie type. III, Manuscripta Math. 85 (1994), no. 2, 195 – 216. · Zbl 0820.20018
[10] Meinolf Geck, On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes, Doc. Math. 1 (1996), No. 15, 293 – 317. · Zbl 0873.20011
[11] Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer, CHEVIE — a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), no. 3, 175 – 210. Computational methods in Lie theory (Essen, 1994). · Zbl 0847.20006
[12] Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. · Zbl 0996.20004
[13] R. Gow, Schur indices of some groups of Lie type, J. Algebra 42 (1976), no. 1, 102 – 120. · Zbl 0352.20013
[14] I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. · Zbl 0337.20005
[15] Gerald J. Janusz, Algebraic number fields, 2nd ed., Graduate Studies in Mathematics, vol. 7, American Mathematical Society, Providence, RI, 1996. · Zbl 0854.11001
[16] N. Kawanaka, Generalized Gel\(^{\prime}\)fand-Graev representations of exceptional simple algebraic groups over a finite field. I, Invent. Math. 84 (1986), no. 3, 575 – 616. · Zbl 0596.20028
[17] G. Lusztig, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976/77), no. 2, 101 – 159. · Zbl 0366.20031
[18] G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), no. 2, 125 – 175. · Zbl 0372.20033
[19] G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), no. 2, 125 – 175. · Zbl 0372.20033
[20] George Lusztig, Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics, vol. 39, American Mathematical Society, Providence, R.I., 1978. Expository lectures from the CBMS Regional Conference held at Madison, Wis., August 8 – 12, 1977. · Zbl 0418.20037
[21] George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. · Zbl 0556.20033
[22] G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205 – 272. · Zbl 0547.20032
[23] George Lusztig, Character sheaves. I, Adv. in Math. 56 (1985), no. 3, 193 – 237. , https://doi.org/10.1016/0001-8708(85)90034-9 George Lusztig, Character sheaves. II, III, Adv. in Math. 57 (1985), no. 3, 226 – 265, 266 – 315. , https://doi.org/10.1016/0001-8708(85)90064-7 George Lusztig, Character sheaves. IV, Adv. in Math. 59 (1986), no. 1, 1 – 63. , https://doi.org/10.1016/0001-8708(86)90036-8 George Lusztig, Character sheaves. V, Adv. in Math. 61 (1986), no. 2, 103 – 155. , https://doi.org/10.1016/0001-8708(86)90071-X George Lusztig, Erratum: ”Character sheaves. V”, Adv. in Math. 62 (1986), no. 3, 313 – 314. , https://doi.org/10.1016/0001-8708(86)90105-2 George Lusztig, Character sheaves. II, III, Adv. in Math. 57 (1985), no. 3, 226 – 265, 266 – 315. , https://doi.org/10.1016/0001-8708(85)90064-7 George Lusztig, Character sheaves. IV, Adv. in Math. 59 (1986), no. 1, 1 – 63. , https://doi.org/10.1016/0001-8708(86)90036-8 George Lusztig, Character sheaves. V, Adv. in Math. 61 (1986), no. 2, 103 – 155. , https://doi.org/10.1016/0001-8708(86)90071-X George Lusztig, Erratum: ”Character sheaves. V”, Adv. in Math. 62 (1986), no. 3, 313 – 314. , https://doi.org/10.1016/0001-8708(86)90105-2 George Lusztig, Character sheaves. II, III, Adv. in Math. 57 (1985), no. 3, 226 – 265, 266 – 315. , https://doi.org/10.1016/0001-8708(85)90064-7 George Lusztig, Character sheaves. IV, Adv. in Math. 59 (1986), no. 1, 1 – 63. , https://doi.org/10.1016/0001-8708(86)90036-8 George Lusztig, Character sheaves. V, Adv. in Math. 61 (1986), no. 2, 103 – 155. , https://doi.org/10.1016/0001-8708(86)90071-X George Lusztig, Erratum: ”Character sheaves. V”, Adv. in Math. 62 (1986), no. 3, 313 – 314. · Zbl 0606.20036
[24] G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 10, 157 – 166. Orbites unipotentes et représentations, I.
[25] George Lusztig, Green functions and character sheaves, Ann. of Math. (2) 131 (1990), no. 2, 355 – 408. · Zbl 0695.20024
[26] George Lusztig, A unipotent support for irreducible representations, Adv. Math. 94 (1992), no. 2, 139 – 179. · Zbl 0789.20042
[27] G. Lusztig, Remarks on computing irreducible characters, J. Amer. Math. Soc. 5 (1992), no. 4, 971 – 986. · Zbl 0773.20011
[28] -, Rationality properties of unipotent representations, J. Algebra 258 (2002), 1-22. · Zbl 1141.20300
[29] Gunter Malle, Die unipotenten Charaktere von ²\?\(_{4}\)(\?²), Comm. Algebra 18 (1990), no. 7, 2361 – 2381 (German). · Zbl 0721.20008
[30] Zyozyu Ohmori, On the Schur indices of certain irreducible characters of reductive groups over finite fields, Osaka J. Math. 25 (1988), no. 1, 149 – 159. · Zbl 0659.20033
[31] Zyozyu Ohmori, On the existence of characters of the Schur index 2 of the simple finite Steinberg groups of type (²\?\(_{6}\)), Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 8, 296 – 298. · Zbl 0827.20009
[32] Zyozyu Ohmori, The Schur indices of the cuspidal unipotent characters of the finite unitary groups, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 6, 111 – 113. · Zbl 0860.20035
[33] Ken-ichi Shinoda, The conjugacy classes of the finite Ree groups of type (\?\(_{4}\)), J. Fac. Sci. Univ. Tokyo Sect. I A Math. 22 (1975), 1 – 15. · Zbl 0306.20014
[34] Toshiaki Shoji, Character sheaves and almost characters of reductive groups. I, II, Adv. Math. 111 (1995), no. 2, 244 – 313, 314 – 354. · Zbl 0832.20065
[35] Toshiaki Shoji, Character sheaves and almost characters of reductive groups. I, II, Adv. Math. 111 (1995), no. 2, 244 – 313, 314 – 354. · Zbl 0832.20065
[36] Toshiaki Shoji, Unipotent characters of finite classical groups, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 373 – 413. · Zbl 0868.20035
[37] G. Lusztig and N. Spaltenstein, On the generalized Springer correspondence for classical groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 289 – 316. N. Spaltenstein, On the generalized Springer correspondence for exceptional groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 317 – 338.
[38] T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167 – 266. · Zbl 0249.20024
[39] Michio Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105 – 145. · Zbl 0106.24702
[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. · Zbl 1051.20008
[41] Alexandre Turull, The Schur indices of the irreducible characters of the special linear groups, J. Algebra 235 (2001), no. 1, 275 – 314. · Zbl 0977.20036
[42] E. WINGS, Über die unipotenten Charaktere der Chevalley-Gruppen vom Typ \(F_4\) in guter Charakteristik, Ph.D. Thesis, RWTH Aachen, 1995. · Zbl 0858.20010
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