ABSTRACT
According to Theorem 4.19, if H is a Banach-Lie subgroup of the real Banach-Lie group G, then G/H carries the structure of a real Banach manifold such that the canonical projection pi : G→ G/H is a real analytic submersion and the natural transitive action
G×G/H → G/H, (g, kH) 7→ gkH is real analytic. In this setting, it is important to know when G/H possesses a G-invariant complex structure. That is, when does G/H possess a structure of complex Banach manifold such that for each g ∈ G, the map
G/H → G/H, kH 7→ gkH, is holomorphic?
