Maximal Compact Subgroups of Lie Groups

This chapter is chiefly devoted to a fundamental theorem which asserts, roughly, that if G is a Lie group with finitely many connected components, then G has maximal compact subgroups, is homeomorphic to the product of any one of them by a euclidean space, and any two maximal compact subgroups of G are conjugate by an inner automorphism. The precise statement, and two corollaries, are given in §1, the proof in §2. The latter makes use of a theorem of Iwasawa on extensions of vector groups by compact groups, and of several results obtained in the previous chapters, notably the decomposition Ad g = _ωK × P, and the conjugacy of maximal compact subgroups in Aut <math display='block'> <mi mathvariant='fraktur'>g</mi> </math>$$\mathfrak{g}$$, where <math display='block'> <mi mathvariant='fraktur'>g</mi> </math>$$\mathfrak{g}$$ is a semisimple Lie algebra. In order to make this chapter more self-contained, we give in §3 alternative proofs of those statements on semi-simple Lie groups, which do not make use of Riemannian geometry, and reproduce a proof of the Iwasawa theorem in §4.