On the irreducibility of Deligne-Lusztig varieties (Q2499724)

Let \(G\) be a connected reductive group defined over an algebraic closure of a finite field and let \(F:G \to G\) be a Frobenius map. This paper proves that the Deligne-Lusztig variety, \(\{gP \in G/P: g^{-1}F(g) \in P \cdot F(P)\}\), associated to a parabolic \(P\) in \(G\), is irreducible if and only if \(P\) is not contained in a proper \(F\)-stable parabolic in \(G\). The proof involves reduction to the case when \(P\) is a Borel subgroup and a geometric argument.