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 ]] ∈ h for all X , Y in h, the complexified group G 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 in [14] 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. A different proof of this generalization of Mostow’s Decomposition Theorem has been obtained independently by G. Larotonda in [13]. Some applications of this Theorem to the geometry of (affine) coadjoint orbits and stable manifolds are given.