Satake-Furstenberg compactifications, the moment map and \lambda_1

Let G be a complex semisimple Lie group, K a maximal compact subgroup and an irreducible representation of K on V. Denote by M the unique closed orbit of G in PP(V) and by OO its image via the moment map. For any measure \mu on M we construct a map from the Satake compactification of G/K (associated to V) to the Lie algebra of K. We call this map, Bourguignon-Li-Yau map since it was introduced by them in the case the complex projective space. If \mu is the K-invariant measure, then the Bourguignon-Li-Yau map is a homeomorphism of the Satake compactification onto the convex envelope of OO. For a large class of measures the image of Bourguignon-Li-Yau map is the convex envelope. As an application we get sharp upper bounds for the first eigenvalue of the Laplacian on functions for a arbitrary K\"ahler metric on a Hermitian symmetric space.