Prehomogeneous spaces for Borel subgroups of general linear groups

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. Further, in case B does not act with a dense orbit, we give the minimal codimension of a B–orbit in n. This can be viewed as a first step towards the difficult open problem of classifying of all ideals n ⊆ u such that B acts on n with a dense orbit. The proofs of our main results require a translation into the representation theory of a certain quasi-hereditary algebra At,1. In this setting we find the minimal dimension of Ext1At,1 (M, M) for a ∆-good At,1–module of certain fixed ∆-dimension vectors.