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Regular elements and Gelfand-Graev representations for disconnected reductive groups. (Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes.) (French) Zbl 1059.20017
Let \(G\) be a connected reductive algebraic group defined over \(\mathbb{F}_q\) and let \(F\) be the corresponding Frobenius endomorphism. The aim is to develop for the disconnected group \(\widetilde G\) the analogue of the Gelfand-Graev representations of the finite group \(G^F\). Here \(\widetilde G\) is obtained from \(G\) by adjoining an automorphism \(\sigma\), satisfying some natural restrictions: It must be rational, it must stabilize a rational Borel group \(B\) and a rational maximal torus \(T\) in \(B\), in its \(\text{Ad}(G)\)-coset the dimension of its centralizer must be maximal.
The main cases are when \(\sigma\) is unipotent or semisimple. In both cases the Gelfand-Graev representations of \(\widetilde G^F\) are constructed in such a manner that they restrict to Gelfand-Graev representations of \(G^F\). Their irreducible constituents have multiplicity one. As in the connected case one defines Harish-Chandra restrictions and shows they send Gelfand-Graev representations to Gelfand-Graev representations. One defines regular conjugacy classes and studies character values on them. But this time those conjugacy classes are in the coset \(G\cdot\sigma\). All this requires a careful study of the similarities and differences between \(G\) and \(\widetilde G\).

MSC:
20C33 Representations of finite groups of Lie type
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
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