SATAKE-FURSTENBERG COMPACTIFICATIONS, THE MOMENT MAP AND λ1

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