Lusztig varieties for regular elements
Xuhua He, Ruben La
TL;DR
This work studies the irreducibility of Lusztig varieties $\mathcal{Y}_{w,h,\delta}$ attached to a Weyl-group element $w$ and a $\delta$-regular element $h$ in a connected reductive group $G$, under the assumption that $\text{supp}_\delta(w)=\mathbb{S}$. It develops equidimensionality results and a fiber-structure analysis, connects geometric properties to the Iwahori–Hecke algebra via rational-point counting over finite fields, and then proves irreducibility of $\mathcal{Y}_{w,h,\delta}$ by reduction to finite fields and cohomological arguments. The main theorem shows irreducibility holds when $w$ has full δ-support and $h$ is $\delta$-regular, generalizing Kim's prior results for regular semisimple and regular unipotent elements. These results provide a robust geometric–representation-theoretic framework with potential applications to affine Lusztig varieties and deeper connections between geometry and character theory.
Abstract
Let $G$ be a connected reductive group over an algebraically closed field. Let $B$ be a Borel subgroup of $G$ 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 $B w B$ with any regular conjugacy class of $G$ is always irreducible. We then prove that the associated Lusztig varieties are irreducible. This extends the previous work of Kim \cite{kim2020homology} on the regular semisimple and regular unipotent elements. The irreducibilitiy result of Lusztig varieties will be used in an upcoming work in the study of affine Lusztig varieties.