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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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