On the canonical real structure on wonderful varieties

TL;DR

This paper investigates the existence and uniqueness of canonical equivariant real structures on spherical and wonderful varieties, providing classifications and orbit estimates for real forms.

Contribution

It introduces the concept of canonical real structures on spherical varieties and proves their existence and uniqueness in key cases, advancing understanding of real forms.

Findings

Existence and uniqueness of canonical structures for certain spherical varieties

Classification of real structures on wonderful embeddings

Estimate of real form orbits on strict wonderful varieties

Abstract

We study equivariant real structures on spherical varieties. We call such a structure canonical if it is equivariant with respect to the involution defining the split real form of the acting reductive group G. We prove the existence and uniqueness of a canonical structure for homogeneous spherical varieties G/H with H self-normalizing and for their wonderful embeddings. For a strict wonderful variety we give an estimate of the number of real form orbits on the set of real points.