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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 |

### Keywords:

rational points; connected reductive group; unipotent radicals; parabolic subgroups; virtual representations; duality operation; orthogonality relations; twisted induction; Levi subgroups
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\textit{P. Deligne} and \textit{G. Lusztig}, J. Algebra 81, 540--545 (1983; Zbl 0535.20020)

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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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