A Bruhat Atlas on the Wonderful Compactification of PSO(2n)/SO(2n-1)

TL;DR

This paper constructs a Bruhat atlas on the wonderful compactification of the symmetric space PSO(2n)/SO(2n-1), linking its stratification to Bruhat cells in type D_{n+1} flag manifolds.

Contribution

It introduces an anticanonical stratification on the compactification and demonstrates the isomorphism of open charts to Bruhat cells, extending Bruhat atlas concepts to this setting.

Findings

Established a stratification compatible with Bruhat cells.

Proved the isomorphism of charts to Bruhat cells in type D_{n+1}.

Provided a new geometric framework for the compactification.

Abstract

A stratified manifold has a Bruhat atlas on it if it can be covered with open charts such that each chart is stratified-isomorphic to an (opposite) Bruhat cell in a (usually Kac-Moody) flag manifold. In this paper, we construct an anticanonical stratification on the wonderful compactification of the symmetric space $PSO(2n)/SO(2n-1)$ and show that the open charts are isomorphic to certain (opposite) Bruhat cells in the type $D_{n+1}$ flag manifold.