Prehomogeneous spaces for Borel subgroups of general linear groups

Abstract

Let k be an algebraically closed field. Let B be the Borel subgroup of GLn(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 At,1. In this setting we find the minimal dimension of Ext 1 A t,1 (M, M ) for a ∆-good At,1-module of certain fixed ∆-dimension vectors.

The orbit structure of Dynkin curves

“…First we consider the case G = SL n (k). We note that in this case the existence of a dense B-orbit in u α can be deduced directly from the main theorem in [12], but we give a more elementary proof here. We take T to be the maximal torus of diagonal matrices in G and B to be the Borel subgroup of upper triangular matrices in G. We write e i,j for the elementary matrix with (i, j)th entry 1 and all other entries 0.…”