2016/06/07 by Goddard, Russell, Goodwin, Simon M.
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1606.02262
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) ∈ \mathfrak p × \mathfrak p | [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.