On commuting varieties of parabolic subalgebras

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.