Mostow's Decomposition Theorem for L*-groups and Applications to affine coadjoint orbits and stable manifolds

Mostow's Decomposition Theorem is a refinement of the polar decomposition. It states the following. Let G be a compact connected semi-simple Lie group with Lie algebra g. Given a subspace h of g such that [X, [X, Y]] belongs to h for all X and Y in h, the complexified group G^C with Lie algebra g + ig is homeomorphic to the product G .exp im. exp ih, where m is the orthogonal of h in g with respect to the Killing form. This Theorem is related to geometric properties of the non-positively curved space of positive-definite symmetric matrices and to a characterization of its geodesic subspaces. The original proof of this Theorem given by Mostow uses the compactness of G. We give a proof of this Theorem using the completeness of the Lie algebra g instead, which can therefore be applied to an L*-group of arbitrary dimension. Some applications of this Theorem to the geometry of stable manifolds and affine coadjoint orbits are given.