Abstract

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

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