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On the classification of \(l\)-blocks of finite groups of Lie type. (English) Zbl 0771.20007

Let \(G\) be a connected reductive group defined over \(\mathbb{F}_ q\) with Frobenius map \(F\) and let \(G^*\) be the dual group of \(G\). For a semisimple \(s\in G^{*F}\) let \({\mathcal E}(G^ F,s)\) denote the set of all \(\chi\in\text{Irr}(G^ F)\) occurring as an irreducible constituent in the generalized Deligne-Lusztig character \(R^ G_{T^*}(s)\). By a result of M. Broué and J. Michel [J. Reine Angew. Math. 395, 56-67 (1989; Zbl 0654.20048)], \[ {\mathcal E}_ l(G^ F,s):= \bigcup_{t}{\mathcal E}(G^ F,st), \] where \(t\) runs through \(C_{G^*}(s)^ F_ l\) and \(s\) is semisimple of order prime to \(l\), is a union of \(l\)-blocks. In the paper under review the author describes in the case of an abelian Sylow-\(l\)-subgroup blocks of maximal defect which are contained in \({\mathcal E}_ l(G^ F,s)\). Under further arithmetical conditions on \(q\) relative to \(l\) he answers Brauer’s height zero conjecture affirmatively for \(l\)-blocks of \(G^ F\). As consequences the author obtains that the regular and semisimple block in \({\mathcal E_ l}(G^ F,s)\) coincide and that for \(l\mid q-1\), the restrictions of the irreducible constituents of \(1^{G^ F}_{B^ F}\) to the \(l'\)-classes form a basic set of the Brauer characters of the principal block.
Reviewer: W.Willems (Mainz)

MSC:

20C20 Modular representations and characters
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C33 Representations of finite groups of Lie type

Citations:

Zbl 0654.20048
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References:

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