Algebraic Geometry On the irreducibility of Deligne–Lusztig varieties

Let G be a connected reductive algebraic group defined over an algebraic closure of a finite field and let F: G → G be an endomorphism such that Fδ is a Frobenius endomorphism for some δ 1. Let P be a parabolic subgroup of G. We prove that the Deligne–Lusztig variety {gP | g−1F(g) ∈ P · F(P)} is irreducible if and only if P is not contained in a proper F-stable paraboli