Commuting Varieties for Nilpotent Radicals

LetUbe the unipotent radical of a Borel subgroup of a connected reductive algebraic groupG, which is defined over an algebraically closed fieldk. In this paper, we extend work by Goodwin and Röhrle concerning the commuting variety of Lie(U) for Char(k) = 0 to fields whose characteristic is good forG.

On Commuting Varieties of Nilradicals of Borel Subalgebras of Reductive Lie Algebras

Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic 0. We consider the commuting variety of the nilradical of the Lie algebra of a Borel subgroup B of G. In case B acts on with only a finite number of orbits, we verify that is equidimensional and that the irreducible components are in correspondence with the distinguishedB-orbits in . We observe that in general is not equidimensional, and determine the irreducible components of in the minimal cases where there are infinitely many B-orbits in .