On commuting varieties of parabolic subalgebras

TL;DR

This paper characterizes when the commuting variety of a parabolic subalgebra is irreducible and normal, linking these properties to the modality of the adjoint action on the nilpotent variety, with classifications for Borel subalgebras.

Contribution

It provides a necessary and sufficient condition for irreducibility of commuting varieties of parabolic subalgebras based on the modality of the adjoint action.

Findings

Characterization of irreducibility of $\

Classification of irreducible commuting varieties for Borel subalgebras.

Conditions for normality of commuting varieties in the irreducible cases.

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $\mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $\mathcal C(\mathfrak p) = \{(X,Y) \in \mathfrak p \times \mathfrak p \mid [X,Y] = 0\}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $\mathcal C(\mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $\mathfrak p$. As a consequence, for the case $P = B$ a Borel subgroup of $G$, we give a classification of when $\mathcal C(\mathfrak b)$ is irreducible; this builds on a partial classification given by Keeton. Further, in cases where $\mathcal C(\mathfrak p)$ is irreducible, we consider whether $\mathcal C(\mathfrak p)$ is a normal variety. In particular, this leads to a classification of when $\mathcal C(\mathfrak b)$ is normal.