Let G be a connected reductive algebraic group defined over an
algebraically closed field \mathbbm k of characteristic zero. We consider the
commuting variety C(u) of the nilradical u of
the Lie algebra b of a Borel subgroup B of G. In case B acts
on u with only a finite number of orbits, we verify that C(u) is equidimensional and that the irreducible components are in
correspondence with the {\em distinguished} B-orbits in u. We
observe that in general C(u) is not equidimensional, and
determine the irreducible components of C(u) in the
minimal cases where there are infinitely many B-orbits in u.Comment: 10 page