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

摘要

Mostow's Decomposition Theorem is a refinement of the polar decomposition. Itstates the following. Let G be a compact connected semi-simple Lie group withLie algebra g. Given a subspace h of g such that [X, [X, Y]] belongs to h forall X and Y in h, the complexified group G^C with Lie algebra g + ig ishomeomorphic to the product G .exp im. exp ih, where m is the orthogonal of hin g with respect to the Killing form. This Theorem is related to geometricproperties of the non-positively curved space of positive-definite symmetricmatrices and to a characterization of its geodesic subspaces. The originalproof of this Theorem given by Mostow uses the compactness of G. We give aproof of this Theorem using the completeness of the Lie algebra g instead,which can therefore be applied to an L*-group of arbitrary dimension. Someapplications of this Theorem to the geometry of stable manifolds and affinecoadjoint orbits are given.