Cartan–Iwasawa–Malcev theorem

Remarks

For semisimple groups with finite center, the result follows from the Cartan decomposition $G = K \exp(\mathfrak{p})$ and from Cartan's fixed point theorem. That theorem says that a compact group acting by isometries on the nonpositively curved space $G/K$ has a fixed point. So each compact subgroup lies in the stabilizer of a point, a conjugate of $K$. The general case is due to Malcev and Iwasawa.

Example: $G = \mathrm{GL}_n^+(\R)$, the invertible real matrices of positive determinant. A maximal compact subgroup is $\mathrm{SO}(n)$, and the polar decomposition gives a homeomorphism of $G$ with $\mathrm{SO}(n) \times \R^{n(n+1)/2}$, the second factor being the symmetric matrices.

The conclusions extend to Lie groups with finitely many connected components, except that $K$ then has one component for each component of $G$. For groups with infinitely many components, maximal compact subgroups may fail to exist.

Going deeper

For a student

The theorem reduces the homotopy type of a connected Lie group to that of a compact Lie group. For example $\mathrm{SL}_2(\R)$ has the homotopy type of a circle, and $\mathrm{SL}_n(\C)$ that of $\mathrm{SU}(n)$, which is simply connected. It also says that the classifying space of $G$ is that of $K$, which is why characteristic classes of $G$-bundles only depend on $K$. Uniqueness up to conjugacy makes $K$ an invariant of $G$, used for example to define the symmetric space $G/K$.