Satake-Furstenberg compactifications, the moment map and \lambda_1

TL;DR

This paper explores the relationship between Satake compactifications, moment maps, and eigenvalue bounds on Hermitian symmetric spaces, providing new geometric insights and sharp eigenvalue estimates.

Contribution

It introduces a map from Satake compactifications to the Lie algebra of K, linking measures on orbits to convex geometry and eigenvalue bounds.

Findings

The map is a homeomorphism for K-invariant measures.

The image of the map is the convex envelope of the orbit for many measures.

Sharp upper bounds for the first Laplacian eigenvalue are derived.

Abstract

Let G be a complex semisimple Lie group, K a maximal compact subgroup and V an irreducible representation of K. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure on M we construct a map from the Satake compactification of G/K (associated to V) to the Lie algebra of K. For the K-invariant measure, this map is a homeomorphism of the Satake compactification onto the convex envelope of O. For a large class of measures the image of the map is the convex envelope. As an application we get sharp upper bounds for the first eigenvalue of the Laplacian on functions for an arbitrary Kaehler metric on a Hermitian symmetric space.