Equivariant models of spherical varieties

TL;DR

This paper investigates conditions under which spherical homogeneous spaces and their embeddings admit equivariant models over subfields, focusing on inner forms of split groups and spherically closed subgroups.

Contribution

It establishes criteria for the existence and uniqueness of $G_0$-equivariant models of spherical varieties under specific algebraic group conditions.

Findings

Existence of $G_0$-models when $H$ is spherically closed

Uniqueness of models when $H$ equals its normalizer

Compatibility of models for embeddings under stronger assumptions

Abstract

Let $G$ be a connected semisimple group over an algebraically closed field $k$ of characteristic 0. Let $Y=G/H$ be a spherical homogeneous space of $G$, and let $Y'$ be a spherical embedding of $Y$. Let $k_0$ be a subfield of $k$. Let $G_0$ be a $k_0$ -model ($k_0$-form) of $G$. We show that if $G_0$ is an inner form of a split group and if the subgroup $H$ of $G$ is spherically closed, then $Y$ admits a $G_0$-equivariant $k_0$-model. If we replace the assumption that $H$ is spherically closed by the stronger assumption that $H$ coincides with its normalizer in $G$, then $Y$ and $Y'$ admit compatible $G_0$-equivariant $k_0$-models, and these models are unique.