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 p be the Lie algebra of P. We consider the commuting variety C(p)={(X,Y)∈p×p|[X,Y]=0}. Our main theorem gives a necessary and sufficient condition for irreducibility of C(p) in terms of the modality of the adjoint action of P on the nilpotent variety of p. As a consequence, for the case P=B a Borel subgroup of G, we give a classification of when C(b) is irreducible; this builds on a partial classification given by Keeton. Further, in cases where C(p) is irreducible, we consider whether C(p) is a normal variety. In particular, this leads to a classification of when C(b) is normal.