Satake-Furstenberg compactifications, the moment map and λ_1

Details Satake-Furstenberg compactifications, the moment map and λ_1 Satake-Furstenberg compactifications, the moment map and λ_1

2010-03-13

arxiv.org/abs/1003.2725v2

Mathematics

Differential Geometry

A few misprints corrected. Reference added. To appear on American Journal of Mathematics

Scientific paper

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.

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