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On unipotent blocks and their ordinary characters. (English) Zbl 0817.20046

Let \(G\) be a connected, reductive algebraic group defined over \(F_ q\), \(F : G \to G\) a Frobenius morphism and \(G^ F\) the finite group of \(F\)- fixed points of \(G\). Let \(\ell\) be an odd prime not dividing \(q\). Assuming that \(\ell\) is a good prime for \(G\) and \(\ell \neq 3\) if \(^ 3 D_ 4\) is involved in \(G\), the authors describe the unipotent \(\ell\)- blocks of \(G\), i.e. the \(\ell\)-blocks containing unipotent characters. Let \(L\) be an \(F\)-stable Levi subgroup of \(G\) and \(d\) a positive integer. Then \(L\) is said to be \(d\)-split if it is the centralizer of a \(\phi_ d\)-subgroup of \(G\) in the sense of M. Broué and G. Malle [Math. Ann. 292, 241-262 (1992)], where \(\phi_ d(x)\) is the \(d\)th cyclotomic polynomial. Let \(R^ G_ L\) and \(^*R^ G_ L\) denote the Lusztig functor and its adjoint. A unipotent \(d\)-pair is a pair \((L,\lambda)\) where \(L\) is a \(d\)-split Levi subgroup of \(G\) and \(\lambda\) is a unipotent character of \(L^ F\), and \((L,\lambda)\) is \(d\)-cuspidal if for every proper \(d\)-split Levi subgroup \(M\) of \(L\), \(^*R^ L_ M(\lambda) = 0\).
The main theorem (4.4) proved by the authors then states that there is a bijection between \(G^ F\)-conjugacy classes of unipotent \(e\)-cuspidal pairs, where \(e\) is the order of \(q \bmod \ell\), and the set of unipotent \(\ell\)-blocks of \(G^ F\). If a block corresponds to a pair \((L,\lambda)\), the unipotent characters in the block are precisely the constituents of \(R^ G_ L(\lambda)\). The defect group of the block is a Sylow \(\ell\)-subgroup of \(C^ 0_ G([L,L])^ F\). Furthermore, all the characters in the block can be described in terms of Lusztig maps \(R^ G_{G(t)}\) where \(t\) is an \(\ell\)-element in the dual group \(G^{*F}\), and \(G(t) \subset G\) is in duality with \(C^ 0_{G*}(t)\). The proof involves an analysis of the centralizers of \(\ell\)-subgroups and \(e\)-split Levi subgroups, and local block theory.
Remark: The theorem was known for special classes of groups, but the authors remark that their proof is more intrinsic. For large \(\ell\), when the defect group is abelian, the general theorem was also proved by M. Broué, G. Malle and J. Michel [Astérisque 212, 7-92 (1993)].

MSC:

20G05 Representation theory for linear algebraic groups
20C20 Modular representations and characters
20G40 Linear algebraic groups over finite fields
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References:

[1] [AB] Alperin, J., Broué, M.: Local methods in block theory. Ann. Math.110, 143-157 (1979) · Zbl 0416.20006 · doi:10.2307/1971248
[2] [BT] Borel, A., Tits, J.: Groupes Réductifs. Publ. Math., Inst. Hautes Étud Sci.27, 55-151 (1965) · Zbl 0145.17402 · doi:10.1007/BF02684375
[3] [B] Brauer, R.: On blocks and sections in finite groups. I. Am. J. Math.89, 1115-1136 (1967) · Zbl 0174.05401 · doi:10.2307/2373422
[4] [Br] Broué, M.: Les ? des groupes GL (n, q) etU(n, q 2) et leurs structures locales. Astérisque133-134, 159-188 (1986)
[5] [BM] Broué, M., Malle, G.: Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis. Math. Ann.292, 241-262 (1992) · Zbl 0820.20057 · doi:10.1007/BF01444619
[6] [BMi] Broué, M., Michel, J.: Blocs et séries de Lusztig dans un groupe réductif fini. J. reine angew. Math.395, 56-67 (1989) · Zbl 0654.20048
[7] [BMM] Broué, M., Malle, G., Michel, J.: Generic blocks of finite reductive groups. Astérisque (to appear) · Zbl 0843.20012
[8] [C] Cabanes, M.: Local structure of thep-blocks of \(\tilde S_n \) . Math. Z.198, 519-543 (1988) · Zbl 0646.20011 · doi:10.1007/BF01162871
[9] [CE] Cabanes, M., Enguehard, M.: Unipotent blocks of finite reductive groups of a given type. Math. Z.213, 479-490 (1993) · Zbl 0795.20021 · doi:10.1007/BF03025733
[10] [Ca1] Carter, R. W.: Conjugacy classes in the Weyl group. Compositio Math.25, 1-59 (1972) · Zbl 0254.17005
[11] [Ca2] Carter, R. W.: Finite Groups of Lie type: Conjugacy classes and Complex Characters. New York, Wiley, 1985
[12] [DM1] Digne, F., Michel, J.: Théorie de Deligne-Lusztig et caractères des groupes linéaires et unitaires. J. of Algebra107, 217-255 (1987) · Zbl 0622.20034 · doi:10.1016/0021-8693(87)90087-1
[13] [DM2] Digne, F., Michel, J.: Representations of finite groups of Lie type. Cambridge: Cambridge University Press, 1991 · Zbl 0815.20014
[14] [DM3] Digne, F., Michel, J.: Groupes Réductifs Non Connexes. Rapp. Rech LMENS92-2, 1992
[15] [FS1] Fong, P., Srinivasan, B.: The blocks of finite general and unitary groups. Invent. Math.69, 109-153 (1982) · Zbl 0507.20007 · doi:10.1007/BF01389188
[16] [FS2] Fong, P. Srinivasan, B.: Generalized Harish-Chandra Theory for Unipotent Characters of Finite Classical Groups. J. of Algebra104, 301-309 (1986 · Zbl 0606.20035 · doi:10.1016/0021-8693(86)90217-6
[17] [FS3] Fong, P., Srinivasan, B.: The blocks of finite classical groups. J. reine angew. Math.396, 122-191 (1989) · Zbl 0656.20039
[18] [GH] Geck, M., Hiss, G.: Basic sets of Brauer characters of finite groups of Lie type. J. reine angew. Math.418, 173-188 (1991) · Zbl 0771.20008
[19] [H] Hiss, G.: Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definie render Charakteristik. Thesis 1990
[20] [L] Lusztig, G.: On the representations of reductive groups with disconnected centre. Astérisque168, 157-166 (1988)
[21] [LS] Lusztig, G., Srinivasan, B.: The characters of the finite unitary groups. J. of Algebra49, 167-171 (1977) · Zbl 0384.20008 · doi:10.1016/0021-8693(77)90277-0
[22] [NT] Nagao, H., Tsushima, Y.: Representations of Finite Groups. New York: Academic, 1989 · Zbl 0673.20002
[23] [P1] Puig, Ll.: The Nakayama conjectures and the Brauer pairs. Pub. Math Univ. Paris VII,25, (1987)
[24] [P2] Puig, Ll.: Algèbres de source de certains blocs des groupes de Chevalley. Astérisque181-182, 221-236 (1990)
[25] [S] Schewe, K.: Blöcke exzeptioneller Chevalley-Gruppen. Bonn. Math. Schr.165, (1985) · Zbl 0644.20027
[26] [SSt] Springer, T., Steinberg, R.: Conjugacy classes. In: Borel A. et al., (eds.) Seminar on algebraic groups and related finite groups. Lect. Notes Math. vol. 131, Berlin Heidelberg New York: Springer (1970) · Zbl 0249.20024
[27] [St] Steinberg, R.: Endomorphisms of linear algebraic groups. Mem.80, (1968) · Zbl 0164.02902
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