Prehomogeneous spaces for Borel subgroups of general linear groups

Abstract

Let k be an algebraically closed field. Let B be the Borel subgroup of GL_n(k) consisting of nonsingular upper triangular matrices. Let b = Lie B 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 A_t,1. In this setting we find the minimal dimension of Ext¹A_t,1(M,M) for a δ-good A_t,1-module of certain fixed δ-dimension vectors.