Heisenberg Equivariant Compactifications of Rational Homogeneous Varieties

Abstract

Let $G/P$ be a complex projective rational homogeneous variety of dimension $2m+1$. We prove that $G/P$ is an equivariant compactification of the Heisenberg group of dimension $2m+1$ if and only if it is isomorphic to either an adjoint variety, or the 3-dimensional smooth quadric $Q^3$, or a product ${\mathbb{P}}^{2m+1-d}\times Y$ with $1\le d=\dim Y\le m$, where $Y$ is a product of cominuscule varieties.