Proofs

For a real semisimple Lie group, Cartan's proof of the existence and uniqueness of a maximal compact subgroup can be found in Borel (1950) and Helgason (1978). Cartier (1955) and Hochschild (1965) discuss the extension to connected Lie groups and connected locally compact groups.

For semisimple groups, existence is a consequence of the existence of a compact real form of the noncompact semisimple Lie group and the corresponding Cartan decomposition. The proof of uniqueness relies on the fact that the corresponding Riemannian symmetric space G/K has negative curvature and Cartan's fixed point theorem. Mostow (1955) showed that the derivative of the exponential map at any point of G/K satisfies |d exp X| ≥ |X|. This implies that G/K is a Hadamard space, i.e. a complete metric space satisfying a weakened form of the parallelogram rule in a Euclidean space. Uniqueness can then be deduced from the Bruhat-Tits fixed point theorem. Indeed, any bounded closed set in a Hadamard space is contained in a unique smallest closed ball, the center of which is called its circumcenter. In particular a compact group acting by isometries must fix the circumcenter of each of its orbits.

Proof of uniqueness for semisimple groups

Mostow (1955) also related the general problem for semisimple groups to the case of GL(n, R). The corresponding symmetric space is the space of positive symmetric matrices. A direct proof of uniqueness relying on elementary properties of this space is given in Hilgert & Neeb (2012).

Let ${\displaystyle {\mathfrak {g}}}$ be a real semisimple Lie algebra with Cartan involution σ. Thus the fixed point subgroup of σ is the maximal compact subgroup K and there is an eigenspace decomposition

${\displaystyle \displaystyle {{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}},}}$

where ${\displaystyle {\mathfrak {k}}}$, the Lie algebra of K, is the +1 eigenspace. The Cartan decomposition gives

${\displaystyle \displaystyle {G=K\cdot \exp {\mathfrak {p}}=K\cdot P=P\cdot K.}}$

If B is the Killing form on ${\displaystyle {\mathfrak {g}}}$ given by B(X,Y) = Tr (ad X)(ad Y), then

${\displaystyle \displaystyle {(X,Y)_{\sigma }=-B(X,\sigma (Y))}}$

is a real inner product on ${\displaystyle {\mathfrak {g}}}$. Under the adjoint representation, K is the subgroup of G that preserves this inner product.

If H is another compact subgroup of G, then averaging the inner product over H with respect to the Haar measure gives an inner product invariant under H. The operators Ad p with p in P are positive symmetric operators. This new inner produst can be written as

${\displaystyle (S\cdot X,Y)_{\sigma },}$

where S is a positive symmetric operator on ${\displaystyle {\mathfrak {g}}}$ such that Ad(h)^tS Ad h = S for h in H (with the transposes computed with respect to the inner product). Moreover, for x in G,

${\displaystyle \displaystyle {\mathrm {Ad} \,\sigma (x)=(\mathrm {Ad} \,(x)^{-1})^{t}.}}$

So for h in H,

${\displaystyle \displaystyle {S\circ \mathrm {Ad} (\sigma (h))=\mathrm {Ad} (h)\circ S.}}$

For X in ${\displaystyle {\mathfrak {p}}}$ define

${\displaystyle \displaystyle {f(e^{X})=\mathrm {Tr} \,\mathrm {Ad} (e^{X})S.}}$

If e_i is an orthonormal basis of eigenvectors for S with Se_i = λ_i e_i, then

${\displaystyle \displaystyle {f(e^{X})=\sum \lambda _{i}(\mathrm {Ad} (e^{X})e_{i},e_{i})_{\sigma }\geq (\min \lambda _{i})\cdot \mathrm {Tr} \,e^{\mathrm {ad} \,X},}}$

so that f is strictly positive and tends to ∞ as |X| tends to ∞. In fact this norm is equivalent to the operator norm on the symmetric operators ad X and each non-zero eigenvalue occurs with its negative, since i ad X is a skew-adjoint operator on the compact real form ${\displaystyle {\mathfrak {k}}\oplus i{\mathfrak {p}}}$.

So f has a global minimum at Y say. This minimum is unique, because if Z were another then

${\displaystyle \displaystyle {e^{Z}=e^{Y/2}e^{X}e^{Y/2},}}$

where X in ${\displaystyle {\mathfrak {p}}}$ is defined by the Cartan decomposition

${\displaystyle \displaystyle {e^{Z/2}e^{-Y/2}=k\cdot e^{X/2}.}}$

If f_i is an orthonormal basis of eigenvectors of ad X with corresponding real eigenvalues μ_i, then

${\displaystyle \displaystyle {g(t)=f(e^{Y/2}e^{tX}e^{Y/2})=\sum e^{\mu _{i}t}\|Ad(e^{Y/2})f_{i}\|_{\sigma }^{2}.}}$

Since the right hand side is a positive combination of exponentials, the real-valued function g is strictly convex if X ≠ 0, so has a unique minimum. On the other hand, it has local minima at t = 0 and t = 1, hence X = 0 and p = exp Y is the unique global minimum. By construction f(x) = f(σ(h)xh⁻¹) for h in H, so that p = σ(h)ph⁻¹ for h in H. Hence σ(h)= php⁻¹. Consequently, if g = exp Y/2, gHg⁻¹ is fixed by σ and therefore lies in K.

Representation theory

Maximal compact subgroups play a basic role in the representation theory when G is not compact. In that case a maximal compact subgroup K is a compact Lie group (since a closed subgroup of a Lie group is a Lie group), for which the theory is easier.

The operations relating the representation theories of G and K are restricting representations from G to K, and inducing representations from K to G, and these are quite well understood; their theory includes that of spherical functions.

Topology

The algebraic topology of the Lie groups is also largely carried by a maximal compact subgroup K. To be precise, a connected Lie group is a topological product (though not a group theoretic product) of a maximal compact K and a Euclidean space – G = K × R^d – thus in particular K is a deformation retract of G, and is homotopy equivalent, and thus they have the same homotopy groups. Indeed, the inclusion ${\displaystyle K\hookrightarrow G}$ and the deformation retraction ${\displaystyle G\twoheadrightarrow K}$ are homotopy equivalences.

For the general linear group, this decomposition is the QR decomposition, and the deformation retraction is the Gram-Schmidt process. For a general semisimple Lie group, the decomposition is the Iwasawa decomposition of G as G = KAN in which K occurs in a product with a contractible subgroup AN.