Spherical orbit closures in simple projective spaces and their normalizations

TL;DR

This paper characterizes the orbit structure and normalization properties of spherical orbit closures in simple projective spaces, providing combinatorial criteria for when the normalization is a homeomorphism.

Contribution

It describes the orbits of spherical orbit closures and their normalizations, and establishes criteria for the normalization to be a homeomorphism in certain cases.

Findings

Orbit descriptions for spherical orbit closures and their normalizations

Criteria for normalization morphism to be a homeomorphism

Conditions satisfied when G is simply laced or H is symmetric

Abstract

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module of finite dimension. If G/H \subset P(V) is a spherical orbit and if X is its closure, then we describe the orbits of X and those of its normalization. If moreover the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup.