The Structure of Real Semisimple Lie Groups

algebra g barycenter BowBo Bruhat decomposition called Cartan decomposition Cartan involution closed subgroup compact real form conjugate convex COROLLARY corresponding cosets Coxeter group deel defined denote diffeomorphism Dynkin diagram ex Q exists finite center follows form of 9 geometry Hausdorff space Hence homeomorphism implies induces invariant subset irreducible ISBN 90 isomorphic Iwasawa decomposition Killing form Lemma Let g Lie incidence system linear mapping morphism morphism of T-spaces normal subgroup notation positive definite PROOF PROPOSITION prove proximal T-space quasi-invariant real Lie algebra real semisimple Lie related to u,t Riemannian symmetric pair root system semisimple Lie algebra semisimple Lie group simply connected strongly proximal subgroup of G Suppose surjective symmetric spaces T-diagram Theorem Tits system topological unique vector space Weyl chambers Weyl group α α α αεφ λι