Character sheaves on nonconnected groups
The author studies character sheaves for nonconnected reductive algebraic groups defined over an algebraic closure \(\overline\mathbb{F}_q\) of a finite field \(\mathbb{F}_q\). Let \(G\) be a connected reductive algebraic group defined over \(\mathbb{F}_q\). We let \(F\colon G\to G\) be the corresponding Frobenius map and \(\sigma\colon G @>\sim>>G\) be a semisimple automorphism, of finite order and commuting with \(F\). The author considers the nonconnected group \(G\rtimes\langle\sigma\rangle\) (or more precisely its connected component \(G\cdot\sigma\)) and proves a character formula for the characteristic functions of perverse sheaves on \(G\cdot\sigma\) which are induced from \(T\)-equivariant tame local systems on \(T\cdot\sigma\) (here \(T\) is a maximal torus of \(G\) which is \(\sigma\)-stable and contained in a \(\sigma\)-stable Borel subgroup of \(G\)). By comparison with the character formula for the generalized Deligne-Lusztig induction functor \(R_{T\cdot\sigma}^{G\cdot\sigma}\) he gets an equality which relates the characteristic functions of induced perverse sheaves to generalized Deligne-Lusztig induction (that equality is analogous to those obtained by Lusztig in case of connected groups). He then shows a partial analog for nonconnected groups of Lusztig's conjecture (that conjecture has been proved by Shoji), which says that the characteristic functions of character sheaves are in an essential way the same as the ``almost characters'' defined by Lusztig. One purpose of the paper under review is also to prove a formula which makes precise the effect of Shintani descent \(\text{Sh}_{F,\sigma^{-1}F}\) on the characteristic functions of \(G\)-equivariant \(F\)-stable perverse sheaves on \(G\cdot\sigma\) (in this part of the paper the automorphism \(\sigma\) is not assumed to be semisimple). Finally the author shows that the functors \(R_{T\cdot\sigma}^{G\cdot\sigma}\) and \(\text{Sh}_{F,\sigma^{-1}F}\) commute.
scientific article; zbMATH DE number 930154