Non-Reductive Conjugation on the Nilpotent Cone

TL;DR

This paper studies the action of certain subgroups of GL(n,C) on nilpotent matrices, providing normal forms, invariants, and insights into algebraic quotients, advancing understanding of conjugation actions in algebraic geometry.

Contribution

It introduces new normal forms and describes generators for invariants under Borel and unipotent subgroup actions on the nilpotent cone, and explores GIT-quotients.

Findings

Derived generic normal forms for orbits.

Described generators for Borel and U-invariant rings.

Initiated study of GIT-quotients for Borel action.

Abstract

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$, especially of the Borel subgroup $B$ and of the standard unipotent subgroup $U$ of the latter on the nilpotent cone of complex nilpotent matrices. We obtain generic normal forms of the orbits and describe generating (semi-) invariants for the Borel semi-invariant ring as well as for the $U$-invariant ring. The latter is described in more detail in terms of algebraic quotients by a special toric variety closely related. The study of a GIT-quotient for the Borel-action is initiated.