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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 \(\widetilde G\).

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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