Groupes réductifs non connexes

Fixed point ratios in actions of finite exceptional groups of Lie type

“…In [18], Digne and Michel developed a Deligne-Lusztig theory for the complex characters of a non-connected reductive group over a finite field. They defined characters which they called generalized Deligne-Lusztig characters; these are extensions of (ordinary) Deligne-Lusztig characters for the relevant connected group.…”

Complements on disconnected reductive groups

“…is a pair of a maximal torus of G 0 and a Borel subgroup of G 0 ; note that a "torus" meets all connected components of G, while (contrary to what is stated erroneously after [DM94,1.4]) this may not be the case for a "Levi".…”

“…Following [DM94,1.15], we call quasi-central a quasi-semi-simple element σ which has maximal dimension of centralizer C G 0 (σ) (that we will also denote by G 0 σ ) amongst all quasi-semi-simple elements of G 0 · σ. In the sequel, we fix a reductive group G and (excepted in the next section where we take a "global" viewpoint) an F -stable connected component G 1 of G. In most of [DM94] we assumed that (G 1 ) F contained a quasi-central element. Here we do not assume this.…”

“…Here we do not assume this. Note however that by [DM94,1.34] G 1 contains an element σ which induces an F -stable quasi-central automorphism of G 0 . Such an element will be enough for our purpose, and we fix one from now on.…”

“…0 is unipotent, and u is F -stable, unipotent and still in N (G 1 ) F (T 0 ⊂ B 0 ) thus quasi-semi-simple, so is quasi-central by [DM94,1.33].…”