Duality for representations of a reductive group over a finite field. II
The first part of this paper appeared in J. Algebra 74, 284-291 (1982; Zbl 0482.20027). The group is the set \(G\) of rational points of a connected reductive group \({\mathcal G}\) over a finite field \(F\); the duality operation has been defined for representations in the first part, from the homology of a complex associated to the fixed vectors of the unipotent radicals of parabolic subgroups defined over \(F\). This second part shows that for the virtual representations introduced by the authors [in Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)] the duality operation is multiplication by a sign which is given. To prove this result, they use some orthogonality relations satisfied by a twisted induction from representations of subgroups of G coming from the Levi subgroups of \({\mathcal G}\) defined over \(F\); the article gives a proof of these orthogonality relations.
scientific article; zbMATH DE number 3849483