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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)

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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