On the irreducibility of Deligne-Lusztig varieties

Cédric Bonnafé (1) , Raphaël Rouquier (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Department of Pure Mathematics

Let $G$ be a connected reductive algebraic group defined over an algebraic closure of a finite field and let $F : G \to G$ be an endomorphism such that $F^d$ is a Frobenius endomorphism for some $d \geq 1$. Let $P$ be a parabolic subgroup of $G$ admitting an $F$-stable Levi subgroup. We prove that the Deligne-Lusztig variety $\{gP~|~g^{-1}F(g)\in P\cdot F(P)\}$ is irreducible if and only if $P$ is not contained in a proper $F$-stable parabolic subgroup of $G$.

Mots clés: Deligne-Lusztig varieties; irreducible components

Domaines: Théorie des groupes [math.GR]

Cédric Bonnafé, Raphaël Rouquier. On the irreducibility of Deligne-Lusztig varieties. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2006, 343, pp.37-39. ⟨hal-00016980v7⟩