Lusztig varieties for regular elements

Abstract

Let $G_0$ be a connected reductive group over an algebraically closed field. Let B be a Borel subgroup of $G_0$ and W be the associated Weyl group. We show that for any $w\in W$ that is not contained in any standard parabolic subgroup of W, the intersection of the Bruhat cell BwB with any regular conjugacy class of $G_0$ is always irreducible. We then prove that the associated Lusztig varieties are irreducible. This extends the previous work of Kim [7] on the regular semisimple and regular unipotent elements. The irreducibility result of Lusztig varieties will be used in an upcoming work in the study of affine Lusztig varieties.