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The characters of the group of rational points of a reductive group with non-connected centre. (English) Zbl 0739.20018
We investigate the complex representation theory of $$G^ F$$ where $$F$$ is a Frobenius endomorphism of a connected reductive group $$G$$ over $$\mathbb{F}_ q$$, whose centre $$Z$$ is not connected. In particular, we study the regular and semisimple characters of $$G^ F$$ and give their values on regular unipotent elements. The Galois cohomology group $$H^ 1(F,Z)$$ enters centrally, as do the geometry of the unipotent classes of $$G$$, the induction and restriction functors $$R^ G_ L$$ and $$^*R^ G_ L$$ of Lusztig and the duality functor. We essentially reduce the character theory of $$G^ F$$ to the solution of problems concerning firstly $$^*R^ G_ L\chi$$ for $$\chi$$ a character of $$G^ F$$ and $$L$$ a rational (not necessarily split) Levi subgroup of $$G$$ and secondly the transition matrix between two sets of unipotently supported class functions on $$G^ F$$ ($$\{\gamma_ u\}$$, $$\{\rho_ v\}$$ see §5).
Reviewer: F.Digne (Amiens)

##### MSC:
 20G05 Representation theory for linear algebraic groups 20G10 Cohomology theory for linear algebraic groups 20G40 Linear algebraic groups over finite fields
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