Satake-Furstenberg compactifications and gradient map

TL;DR

This paper explores the geometric and representation-theoretic structures underlying Satake-Furstenberg compactifications, revealing how the facial structure of a polar orbitope encodes information about parabolic subgroups and providing new insights into compactification descriptions.

Contribution

It establishes a novel connection between the facial structure of a polar orbitope and the classification of irreducible representations of parabolic subgroups, extending results to complex reductive cases.

Findings

The set of irreducible representations is determined by the facial structure of the orbitope.

Each parabolic subgroup has a unique well-adapted closed orbit.

The geometric description of Satake compactifications is achieved without root data.

Abstract

Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak g=\mathfrak k \oplus \mathfrak p$ be a Cartan decomposition of its Lie algebra. Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak k$ and let $\tau$ be an irreducible representation of $G$ on a complex vector space $V$. Let $h$ be a Hermitian scalar product on $V$ such that $\tau(G)$ is compatible with respect to $\mathrm{U}(V,h)^{\mathbb C}$. We denote by $\mu_{\mathfrak p}:\mathbb P(V) \longrightarrow \mathfrak p$ the $G$-gradient map and by $\mathcal O$ the unique closed orbit of $G$ in $\mathbb P(V)$, which is a $K$-orbit, contained in the unique closed orbit of the Zariski closure of $\tau(G)$ in $\mathrm{U}(V,h)^{\mathbb C}$. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of $G$ induced by $\tau$ are completely determined by the facial structure of the polar orbitope $\mathcal E=\mathrm{conv}(\mu_{\mathfrak p} (\mathcal O))$. Moreover, any parabolic subgroup of $G$ admits a unique closed orbit which is well-adapted to $\mathcal O$ and $\mu_{\mathfrak p}$ respectively. These results are new also in the complex reductive case. The connection between $\mathcal E$ and $\tau$ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure $\gamma$ on $\mathcal O$, we construct a map $\Psi_\gamma$ from the Satake compactification of $G/K$ associated to $\tau$ and $\mathcal E$. If $\gamma$ is a $K$-invariant measure then $\Psi_\gamma$ is an homeomorphism of the Satake compactification and $\mathcal E$. Finally, we prove that for a large class of measures the map $\Psi_\gamma$ is surjective.