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 (Figure presented.) of characteristic 0. We consider the commuting variety C((Figure presented.)) of the nilradical (Figure presented.) of the Lie algebra (Figure presented.) of a Borel subgroup B of G. In case B acts on (Figure presented.) with only a finite number of orbits, we verify that C((Figure presented.)) is equidimensional and that the irreducible components are in correspondence with the distinguished B-orbits in (Figure presented.). We observe that in general C((Figure presented.)) is not equidimensional, and determine the irreducible components of C((Figure presented.)) in the minimal cases where there are infinitely many B-orbits in (Figure presented.).