Geometric Conditions for Twisted O-Sphericity

TL;DR

This paper generalizes the geometric conditions for spherical varieties by incorporating a character twist, extending previous definitions and demonstrating the broader applicability through an example.

Contribution

It introduces a twisted version of the spherical variety condition and proves its equivalence to a generalized group action property, expanding the theoretical framework.

Findings

Established the equivalence between twisted spherical conditions and group action properties.

Provided an example illustrating the necessity of the generalization.

Extended the geometric characterization of spherical varieties.

Abstract

The geometric condition defining a spherical variety for a reductive algebraic group was generalized in [AG21], with applications to representation theory. We twist by a character to generalize this definition, and show its equivalence to a property of group actions that generalizes Theorem B of [AG21]. We also present an example to demonstrate the necessity of this generalization.