---
tag: C846XS3Y0T
kind: statement
role: theorem
name: KAK decomposition
msc: ["22E46", "22E15"]
uses: [XJS07AJ50Y, A5JB9JQKR0, H3QX1ZFC8K]
about: [XJS07AJ50Y]
sources:
  - url: https://en.wikipedia.org/wiki/Cartan_decomposition
---

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center and Lie algebra $\mathfrak{g}$.
Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be a Cartan decomposition, as in the [[Cartan decomposition of a Lie group|A5JB9JQKR0]].
Let $K$ be the connected subgroup with Lie algebra $\mathfrak{k}$.
Let $\mathfrak{a} \subseteq \mathfrak{p}$ be a maximal abelian subspace, and let $A = \exp \mathfrak{a}$.
Let $N_K(\mathfrak{a})$ and $Z_K(\mathfrak{a})$ be the normalizer and the centralizer of $\mathfrak{a}$ in $K$ under the adjoint action.
Let $W = N_K(\mathfrak{a}) / Z_K(\mathfrak{a})$.
Then $G = KAK$: each $g \in G$ can be written $g = k_1 \exp(H) k_2$ with $k_1, k_2 \in K$ and $H \in \mathfrak{a}$.
In such a product, $H$ is unique up to the action of $W$ on $\mathfrak{a}$.

## 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 $k \in K$.

## Explanations

### L1 human

Every element of a semisimple Lie group is a rotation, then a stretching along fixed axes, then another rotation.
For matrices this is the singular value decomposition.
The amounts of stretching are well defined up to the order of the axes.

### L2 human

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.