On commuting varieties of parabolic subalgebras

Russell Goddard, Simon M. Goodwin

Published 2016-06-07Version 1

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 we give a classification of when $\mathcal C(\mathfrak b)$ is irreducible for the case $P = B$ a Borel subgroup of $G$; 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.