Satake-Furstenberg compactifications, the moment map and λ1

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 P(V) and by 0 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 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 Kahler metric on a Hermitian symmetric space