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Satake-Furstenberg compactifications, the moment map and λ1
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 1, February 2013
- pp. 237-274
- 10.1353/ajm.2013.0006
- Article
- Additional Information
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.