Parabolic Conjugation and Commuting Varieties

TL;DR

This paper studies the action of upper-block parabolic subgroups on nilpotent matrices, providing a classification of when finitely many orbits occur, with applications to commuting varieties and nested punctual Hilbert schemes.

Contribution

It introduces a finiteness criterion for the number of orbits of parabolic subgroup actions on nilpotent matrices, linking Lie theory and representation theory.

Findings

Classifies actions with finitely many orbits over infinite fields

Establishes a new connection between parabolic actions and representation theory

Applies results to commuting varieties and nested punctual Hilbert schemes

Abstract

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of the general linear group on the variety of nilpotent matrices in its Lie algebra. Lie-theoretically, it is natural to wonder about the number of orbits of this action. We translate the setup to a representation-theoretic one and obtain a finiteness criterion which classifies all actions with only a finite number of orbits over an arbitrary infinite field. These results are applied to commuting varieties and nested punctual Hilbert schemes.