Height 0 characters of finite groups of Lie type

Malle, Gunter

doi:10.1090/S1088-4165-07-00312-3

Height 0 characters of finite groups of Lie type

Author: Gunter Malle
Journal: Represent. Theory 11 (2007), 192-220
MSC (2000): Primary 20C33, 20G40
DOI: https://doi.org/10.1090/S1088-4165-07-00312-3
Published electronically: December 5, 2007
MathSciNet review: 2365640
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a classification of irreducible characters of finite groups of Lie type of $p’$-degree, where $p$ 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 $p$-subgroup. The McKay conjecture asserts that there exists a bijection between characters of $p’$-degree of a group and of the normalizer of a Sylow $p$-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.

References

Michel Broué and Gunter Malle, Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis, Math. Ann. 292 (1992), no. 2, 241–262 (French). MR 1149033, DOI https://doi.org/10.1007/BF01444619
Michel Broué and Gunter Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189 (German). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235834
Michel Broué and Gunter Malle, Generalized Harish-Chandra theory, Representations of reductive groups, Publ. Newton Inst., vol. 16, Cambridge Univ. Press, Cambridge, 1998, pp. 85–103. MR 1714151, DOI https://doi.org/10.1017/CBO9780511600623.006
Michel Broué, Gunter Malle, and Jean Michel, Generic blocks of finite reductive groups, Astérisque 212 (1993), 7–92. Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235832
M. Broué, G. Malle, and J. Michel, Towards spetses. I, Transform. Groups 4 (1999), no. 2-3, 157–218. Dedicated to the memory of Claude Chevalley. MR 1712862, DOI https://doi.org/10.1007/BF01237357
Michel Broué and Jean Michel, Blocs à groupes de défaut abéliens des groupes réductifs finis, Astérisque 212 (1993), 93–117 (French). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235833
Marc Cabanes, Unicité du sous-groupe abélien distingué maximal dans certains sous-groupes de Sylow, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 10, 889–894 (French, with English and French summaries). MR 1278146
Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
François Digne and Jean Michel, On Lusztig’s parametrization of characters of finite groups of Lie type, Astérisque 181-182 (1990), 6, 113–156 (English, with French summary). MR 1051245
François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841
François Digne and Jean Michel, Groupes réductifs non connexes, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 3, 345–406 (French, with English and French summaries). MR 1272294
Meinolf Geck and Gerhard Hiss, Basic sets of Brauer characters of finite groups of Lie type, J. Reine Angew. Math. 418 (1991), 173–188. MR 1111205
Daniel Gorenstein and Richard Lyons, The local structure of finite groups of characteristic $2$ type, Mem. Amer. Math. Soc. 42 (1983), no. 276, vii+731. MR 690900, DOI https://doi.org/10.1090/memo/0276
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592

M. Isaacs, G. Malle, G. Navarro, A reduction theorem for the McKay conjecture. Invent. Math. 170 (2007), 33–101.

Peter B. Kleidman, The maximal subgroups of the Steinberg triality groups $^3D_4(q)$ and of their automorphism groups, J. Algebra 115 (1988), no. 1, 182–199. MR 937609, DOI https://doi.org/10.1016/0021-8693%2888%2990290-6

G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 10, 157–166. Orbites unipotentes et représentations, I. MR 1021495

Gunter Malle, Die unipotenten Charaktere von ${}^2F_4(q^2)$, Comm. Algebra 18 (1990), no. 7, 2361–2381 (German). MR 1063140, DOI https://doi.org/10.1080/00927879008824026

Gunter Malle, Darstellungstheoretische Methoden bei der Realisierung einfacher Gruppen vom Lie Typ als Galoisgruppen, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 443–459 (German). MR 1112174, DOI https://doi.org/10.1007/978-3-0348-8658-1_20

Gunter Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra 177 (1995), no. 3, 768–826 (German, with German summary). MR 1358486, DOI https://doi.org/10.1006/jabr.1995.1329

Gunter Malle, Degrés relatifs des algèbres cyclotomiques associées aux groupes de réflexions complexes de dimension deux, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 311–332 (French). MR 1429878

Gunter Malle, On the generic degrees of cyclotomic algebras, Represent. Theory 4 (2000), 342–369. MR 1773866, DOI https://doi.org/10.1090/S1088-4165-00-00088-1

G. Malle, The inductive McKay condition for simple groups not of Lie type, to appear in Comm. Algebra.

B. Späth, Die McKay-Vermutung für quasi-einfache Gruppen vom Lie-Typ. Dissertation, TU Kaiserslautern, 2007.

B. Späth, The McKay-conjecture for exceptional groups and odd primes, submitted, 2007.

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. MR 0268192

Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335

Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728

Robert Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92. MR 354892, DOI https://doi.org/10.1016/0001-8708%2875%2990125-5

Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575