Characterizing quasi-affine spherical varieties via the automorphism group

TL;DR

This paper shows that for quasi-affine G-spherical varieties, the weight monoid can be characterized by certain G_a-actions, and that smooth affine G-spherical varieties are uniquely determined by their automorphism groups.

Contribution

It establishes a novel link between the weight monoid and G_a-actions, and demonstrates that automorphism groups uniquely identify certain smooth affine G-spherical varieties.

Findings

Weight monoid determined by G_a-actions with respect to a Borel subgroup.

Smooth affine G-spherical varieties (not tori) are characterized by their automorphism groups.

Provides a new method to classify G-spherical varieties using automorphism groups.

Abstract

Let $G$ be a connected reductive algebraic group. In this note we prove that for a quasi-affine $G$-spherical variety the weight monoid is determined by the weights of its non-trivial $\mathbb{G}_a$-actions that are homogeneous with respect to a Borel subgroup of $G$. As an application we get that a smooth affine $G$-spherical variety that is non-isomorphic to a torus is determined by its automorphism group inside the category of smooth affine irreducible varieties.