Equivariant compactifications of a unipotent group by a smooth projective horospherical variety of Picard number one

RESUMO

Cheong proved that if $S$ is a simple Lie group and $P$ is a parabolic subgroup, then $S/P$ admits a unique equivariant compactification of the unipotent radical $N$ of $P$, up to isomorphism, provided that $S/P$ is not isomorphic to a projective space. We generalize this result to a smooth projective horospherical variety $X$ with Picard number 1. Let $G = \mathrm{Aut}(X)$, and let $H$ be the isotropy subgroup at a point in the open $G$-orbit. We describe a unipotent subgroup $N$ of $H$, which is not necessarily the unipotent radical of $H$ when $X$ is not homogeneous. Moreover, we prove that $X$ admits a unique equivariant compactification of $N$, up to isomorphism.