The quotient of a complete symmetric variety

TL;DR

This paper investigates the quotient of a completed symmetric variety by a subgroup, showing it is isomorphic to a closure related to an isotropic torus and describing semistable points in smooth cases.

Contribution

It establishes a new isomorphism for the quotient of symmetric varieties and characterizes semistable points in smooth, toroidal completions.

Findings

Quotient is isomorphic to the closure of an isotropic torus image.

Describes semistable points in smooth, toroidal cases.

Provides structural insights into symmetric varieties and their quotients.

Abstract

We study the quotient of a completion of a symmetric variety G/H under the action of H. We prove that this is isomorphic to the closure of the image of an isotropic torus under the action of the restricted Weyl group. In the case the completion is smooth and toroidal we describe the set of semistable points.