Prescribing Ricci Curvature on Complexified Symmetric Spaces

The complexification of the compact group G is the group G whose Lie algebra is the complexification of the Lie algebra g of G and which satisfies π1(G ) = π1(G). The complexification G/K of G/K can be then identified (G-equivariantly) with the tangent bundle of G/K. We also remark that the Kähler form obtained in the Theorem is exact. This result has been proved in [9] for symmetric spaces of rank 1 and in [2] for compact groups, i.e. for the case when G = K × K and K acts diagonally. For hermitian symmetric spaces and ρ = 0, Theorem 1 has also been known [6, 4]. The proof given here is quite different from that given for group manifolds in [2]. We show that the complex Monge-Ampère equation on G/K reduces, for Ginvariant functions, to a real Monge-Ampère equation on the dual symmetric space G/K. We also show that the Monge-Ampère operator on non-compact symmetric spaces has a radial part, i.e. it is equal, for K-invariant functions, to another Monge-Ampère operator on the maximal abelian subspace of G/K. These facts, together with the theorem on K-invariant real Monge-Ampère equations proved in [3], yield Theorem 1.