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

Satake-Furstenberg compactifications, the moment map and \lambda_1