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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:

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