Let k be an algebraically closed field. Let B be the Borel subgroup of GLn(k) consisting of nonsingular upper triangular matrices. Let b = LieB be the Lie algebra of upper triangular n × n matrices and u the Lie subalgebra of b consisting of strictly upper triangular matrices. We classify all Lie ideals n of b, satisfying u ⊆ n ⊆ u, such that B acts (by conjugation) on n with a dense orbit. Fur...

Let G be a reductive linear algebraic group, P a parabolic subgroup of G and Pu its unipotent radical. We consider the adjoint action of P on the Lie algebra pu of Pu. Richardson’s dense orbit theorem says that there is a dense P -orbit in pu. We consider some instances when P acts with a dense orbit on terms p u of the descending central series of pu. In particular, we show (in good characteri...