Complements on disconnected reductive groups

Abstract

Abstract. We present various results on disconnected reductive groups, in particular about the characteristic 0 representation theory of such groups over finite fields.

Derived categories and Deligne-Lusztig varietiesII

“…-Let G be a (possibly disconnected) reductive algebraic group endowed with an endomorphism F , a power F δ of which is a Frobenius endomorphism defining a rational structure over a finite field q of characteristic p . We refer to [DigMi2,DigMi3] for basic results on disconnected groups.…”

“…We consider now again a non-necessarily connected reductive group G. We fix an element g ∈ G which stabilizes a pair (T, B) where B is a Borel subgroup of G and T is a maximal torus of B. Such an element is called quasi-semisimple in [DigMi2] and [DigMi3]. For instance, any semisimple element of G is quasi-…”

“…This does generalize to disconnected groups, as we explain below. A direct approach along the lines of Broué-Michel is possible, based on the results of [DigMi3]. While we will not use the results in the remaining part of this section, they might be useful for character theoretic questions.…”

Quasi-semisimple elements

“…Each of the signs determines a root in Σ tσ colinear to ᾱ: with our choice of σ (see Context 2.1) in case A 2r this root is π(α) if ᾱ(t) = −1, and π(α + σ i (α)) = 2π(α) otherwise, where 2i = |O(α)|. Thus by [DM15,remark after 8.5], the root systems Φ tσ and Σ tσ have same Weyl group, which is W 0 (tσ) by Proposition 1.11. Finally…”

“…Each of the signs determines a root in colinear to : with our choice of (see Context ) in case this root is if , and otherwise, where . Thus by [, remark after 8.5], the root systems and have same Weyl group, which is by Proposition . Finally thus is equivalent to .…”