KAK decomposition

Remarks

The idea of the proof: by the Cartan decomposition, $g = k \exp X$ with $X \in \mathfrak{p}$. Each element of $\mathfrak{p}$ is conjugate under $\mathrm{Ad}(K)$ to an element of $\mathfrak{a}$, so $X = \mathrm{Ad}(k_2^{-1}) H$ and $\exp X = k_2^{-1} \exp(H) k_2$. Uniqueness comes from the fact that two elements of $\mathfrak{a}$ conjugate under $\mathrm{Ad}(K)$ are conjugate under the Weyl group $W$.

Example: $G = \mathrm{SL}_n(\R)$, $K = \mathrm{SO}(n)$, $A$ the positive diagonal matrices of determinant $1$. The theorem is the singular value decomposition $g = k_1 \operatorname{diag}(\sigma_1, \dots, \sigma_n) k_2$. Here $W$ is the symmetric group $S_n$, and the singular values are unique up to order.

The factors $k_1$ and $k_2$ are not unique: for $g = e$ one may take $k_1 = k$, $H = 0$, $k_2 = k^{-1}$ for any $k \in K$.

Going deeper

For a student

The decomposition is the group form of the fact that $K$ acts on the symmetric space $G/K$ and each orbit meets the flat $A \cdot o$. It gives polar coordinates on $G$. In these coordinates the Haar measure has a density in $H$, a product of powers of $\abs{\sinh \lambda(H)}$ over the positive restricted roots $\lambda$. This is the starting point of harmonic analysis on $G/K$ and of the study of spherical functions.