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Duality for representations of a reductive group over a finite field. II. (English) Zbl 0535.20020
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.
Reviewer: P.Gérardin

##### MSC:
 20G05 Representation theory for linear algebraic groups 20G40 Linear algebraic groups over finite fields
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##### References:
 [1] Deligne, P; Lusztig, G, Representations of reductive groups over finite fields, Ann. of math., 103, 103-161, (1976), (2) · Zbl 0336.20029 [2] Deligne, P; Lusztig, G, Duality for representations of a reductive group over a finite field, J. algebra, 74, 284-291, (1982) · Zbl 0482.20027 [3] Lusztig, G, On the finiteness of the number of unipotent classes, Invent. math., 34, 201-213, (1976) · Zbl 0371.20039
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