Equivariant compactifications of a nilpotent group by $G/P$

TL;DR

This paper studies the unique ways a nilpotent group can be compactified within a flag variety, showing that such compactifications are mostly unique except for projective spaces.

Contribution

It proves the uniqueness of equivariant compactifications of a nilpotent group on flag varieties, except in the case of projective spaces.

Findings

Uniqueness of equivariant compactifications for most cases.

Exception of projective spaces where multiple compactifications exist.

Clarification of the structure of compactifications in algebraic group actions.

Abstract

Let $G$ be a simple complex algebraic group, $P$ a parabolic subgroup of $G$ and $N$ the unipotent radical of $P.$ The so-called equivariant compactification of $N$ by $G/P$ is given by an action of $N$ on $G/P$ with a dense open orbit isomorphic to $N$. In this article, we investigate how many such equivariant compactifications there exist. Our result says that there is a unique equivariant compactification of $N$ by $G/P$, up to isomorphism, except $\P^n$.