On the irreducibility of Deligne-Lusztig varieties

Details On the irreducibility of Deligne-Lusztig varieties On the irreducibility of Deligne-Lusztig varieties

2006-01-16

arxiv.org/abs/math/0601373v5

Comptes Rendus de l Acad\'emie des Sciences - Series I - Mathematics 343 (2006) 37-39

Mathematics

Group Theory

4 pages

Scientific paper

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 $G$.

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