Quiver-graded Richardson orbits

Abstract

In Lie theory, a dense orbit in the nilpotent radical of a parabolic group under the operation of the parabolic is called a Richardson orbit. We define a quiver-graded version of Richardson orbits generalizing the classical definition in the case of the general linear group. We define a quasi-hereditary algebra called the nilpotent quiver algebra whose isomorphism classes of Δ-filtered modules correspond to orbits in our generalized setting. We translate the existence of a Richardson orbit into the existence of a rigid Δ-filtered module of a given dimension vector. We study an idempotent recollement of this algebra whose associated intermediate extension functor can be used to produce Richardson orbits in some situations. This can be explicitly calculated in examples. We also give examples where no Richardson orbit exists.