Invariant deformations of orbit closures in $\mathfrak{sl}_n$

TL;DR

This paper investigates how orbit closures in the Lie algebra of the special linear group deform, revealing that their invariant Hilbert schemes correspond to affine space quotients of sheets, providing explicit geometric descriptions.

Contribution

It establishes that for the special linear group, the invariant Hilbert schemes' connected components are affine space quotients of sheets, linking deformation theory with explicit geometric structures.

Findings

Invariant Hilbert schemes correspond to affine space quotients of sheets.

Connected components of these schemes are geometric quotients.

Results extend Katsylo's construction to the special linear group case.

Abstract

We study deformations of orbit closures for the action of a connected semisimple group $G$ on its Lie algebra $\mathfrak{g}$, especially when $G$ is the special linear group. The tools we use are on the one hand the invariant Hilbert scheme and on the other hand the sheets of $\mathfrak{g}$. We show that when $G$ is the special linear group, the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of $\mathfrak{g}$. These quotients were constructed by Katsylo for a general semisimple Lie algebra $\mathfrak{g}$; in our case, they happen to be affine spaces.