Simple linear compactifications of odd orthogonal groups

TL;DR

This paper classifies simple linear compactifications of odd orthogonal groups, specifically SO(2r+1), by analyzing orbit closures in projective spaces derived from rational modules.

Contribution

It provides a complete classification of such compactifications, highlighting their structure and properties for the first time.

Findings

Classification of simple linear compactifications of SO(2r+1)

Description of orbit closures in projective spaces

Identification of unique closed orbits

Abstract

We classify the simple linear compactifications of SO(2r+1), namely those compactifications with a unique closed orbit which are obtained by taking the closure of the SO(2r+1)xSO(2r+1)-orbit of the identity in a projective space P(End(V)), where V is a finite dimensional rational SO(2r+1)-module.