Affine embeddings of a reductive group

TL;DR

This paper classifies affine varieties with reductive group actions where the group forms an open orbit, linking these varieties to affine embeddings of the group through associated one-parameter subgroups.

Contribution

It introduces a classification method for affine varieties with reductive group actions based on associated one-parameter subgroups, connecting to affine embeddings of the group.

Findings

Classification of affine varieties with reductive group actions

Characterization of such varieties via one-parameter subgroups

Applications to the existence of morphisms

Abstract

We classify affine varieties with an action of a connected, reductive algebraic group such that the group is isomorphic to an open orbit in the variety. This is accomplished by associating a set of one-parameter subgroups of the group to the variety, characterizing such sets, and proving that sets of this type correspond to affine embeddings of the group. Applications of this classification to the existence of morphisms are then given.