Some Applications of Plurisubharmonic Functions to Orbits of Real Reductive Groups

Theorem 1. Let G ⊂ GL(n,R) be a real reductive group with Cartan decomposition g = k⊕ p, g being the Lie algebra of G. Let GC be the subgroup of GL(n,C) with Lie algebra g ⊕ ig and K̃ the subgroup of GC whose Lie algebra is k⊕ ip. Let ΩC be a complex homogeneous space for GC and φ a K̃-invariant strictly plurisubharmonic function on ΩC. If Ω is a G-orbit in ΩC and f = φ|Ω has a critical point, then f is proper, Ω is closed in ΩC and the critical set of f is a single K-orbit, K being the subgroup of G whose Lie algebra is k. Moreover, the function f achieves its minimum value on its critical set.