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Non-connected reductive groups. (Groupes réductifs non connexes.) (French) Zbl 0846.20040

Let \(G\) be a non-connected reductive algebraic group defined over a finite field \(\mathbb{F}_q\), with Frobenius map \(F\). The authors develop a Deligne-Lusztig theory for the complex characters of \(G^F\). They base their investigations on the papers of Malle, Spaltenstein, Steinberg and others [cf. P. Deligne and G. Lusztig, Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029); G. Malle, J. Algebra 159, No. 1, 64-97 (1993; Zbl 0812.20024); N. Spaltenstein, Classes unipotentes et sous-groupes de Borel (Lect. Notes Math. 946, 1982; Zbl 0486.20025); R. Steinberg, Mem. Am. Math. Soc. 80 (1968; Zbl 0164.02902)].

MSC:

20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
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References:

[1] T. ASAI , The unipotent class functions of exceptional groups over finite fields (Comm. in Algebra, Vol. 22, n^\circ 12, 1984 , pp. 2729-2857). MR 86a:20046 | Zbl 0547.20037 · Zbl 0547.20037 · doi:10.1080/00927878408823130
[2] M. BROUÉ , G. MALLE et J. MICHEL , Generic blocks of finite reductive groups (Astérisque, Vol. 212, 1993 , pp. 7-92). MR 95d:20072 | Zbl 0843.20012 · Zbl 0843.20012
[3] P. DELIGNE et G. LUSZTIG , Representations of reductive groups over finite fields (Annals of Mathematics, n^\circ 103, 1976 , pp. 103-161). MR 52 #14076 | Zbl 0336.20029 · Zbl 0336.20029 · doi:10.2307/1971021
[4] F. DIGNE , Shintani-Descent and L Functions on Deligne-Lusztig Varieties (Proceedings of Symposia in Pure Mathematics, Vol. 47, part 1, pp. 61-68). MR 89b:20083 | Zbl 0657.20036 · Zbl 0657.20036
[5] F. DIGNE et J. MICHEL , Fonctions L des variétés de Deligne-Lusztig et descente de Shintani (Mémoires de la Société Mathématique de France, n^\circ 20, 1985 ). Numdam | MR 87h:20071 | Zbl 0608.20027 · Zbl 0608.20027
[6] F. DIGNE et J. MICHEL , Representation of finite groups of Lie type (London Math. Soc. Student Texts, n^\circ 21, Cambridge University Press, 1991 ). MR 92g:20063 | Zbl 0815.20014 · Zbl 0815.20014
[7] F. DIGNE et J. MICHEL , Lusztig Functor and Shintani Descent (Rapports de Recherche du L.M.E.N.S., n^\circ 2, Publications du D.M.I., E.N.S., Paris, 1989 ).
[8] J. E. HUMPHREYS , Linear algebraic groups (Graduate Texts in Mathematics, n^\circ 21, Springer 1975 ). MR 53 #633 | Zbl 0325.20039 · Zbl 0325.20039
[9] G. LUSZTIG , Characters of reductive groups over a finite field (Annals of math. studies, n^\circ 107, Princeton University Press, 1984 ). MR 86j:20038 | Zbl 0556.20033 · Zbl 0556.20033
[10] G. MALLE , Generalized Deligne-Lusztig characters (Journal of Algebra, Vol. 159, n^\circ 1, 1993 , pp. 64-97). MR 94i:20025 | Zbl 0812.20024 · Zbl 0812.20024 · doi:10.1006/jabr.1993.1147
[11] T. SHOJI , Shintani descent for exceptional groups over a finite field (J. Fac. Sci. Uni. Tokyo Sect. 1A, n^\circ 34, 1987 , pp. 599-653). MR 88m:20086 | Zbl 0643.20019 · Zbl 0643.20019
[12] N. SPALTENSTEIN , Classes unipotentes et sous-groupes de Borel (Lecture Notes in Mathematics, n^\circ 946, Springer 1982 ). MR 84a:14024 | Zbl 0486.20025 · Zbl 0486.20025
[13] R. STEINBERG , Endomorphisms of linear algebraic groups (Memoirs of AMS, n^\circ 80, 1968 ). MR 37 #6288 | Zbl 0164.02902 · Zbl 0164.02902
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