Reductive and unipotent actions of affine groups

TL;DR

This paper generalizes classical geometric invariant theory to affine algebraic groups acting on affine varieties, introducing new notions of reductive and unipotent actions and establishing foundational results in this broader context.

Contribution

It extends classical GIT concepts to a relative setting with affine groups and varieties, defining reductive and unipotent actions via fixed point functors and proving key theorems.

Findings

Existence of quotients in the generalized setting

Finite generation of invariants for affine group actions

Partial geometric description of reductive actions

Abstract

We present a generalized version of classical geometric invariant theory \`a la Mumford where we consider an affine algebraic group $G$ acting on a specific affine algebraic variety $X$. We define the notions of linearly reductive and of unipotent action in terms of the $G$ fixed point functor in the category of $(G,\Bbbk [X])$--modules. In the case that $X=\{\star\}$ we recuperate the concept of lineraly reductive and of unipotent group. We prove in our "relative" context some of the classical results of GIT such as: existence of quotients, finite generation of invariants, Kostant--Rosenlicht's theorem and Matsushima's criterion. We also present a partial description of the geometry of such linearly reductive actions.