Torus Actions on Quotients of Affine Spaces

TL;DR

This paper investigates the fixed point loci of torus actions on GIT quotients of complex vector spaces, revealing they are GIT quotients of linear subspaces by Levi subgroups under certain conditions.

Contribution

It demonstrates that fixed point components of torus actions on GIT quotients are themselves GIT quotients of linear subspaces by Levi subgroups, assuming free action on the stable locus.

Findings

Fixed point loci are GIT quotients of linear subspaces.

Components are related to Levi subgroups.

Results depend on free action on the stable locus.

Abstract

We study the locus of fixed points of a torus action on a GIT quotient of a complex vector space by a reductive complex algebraic group which acts linearly. We show that, under the assumption that $G$ acts freely on the stable locus, the components of the fixed point locus are again GIT quotients of linear subspaces by Levi subgroups.