Global Cartan decomposition - Imported Knowlpedia Content

Statement

Let $G$ be a real reductive Lie group, let $K$ be the fixed subgroup of a global Cartan involution, and let $\mathfrak g=\mathfrak k\oplus\mathfrak p$ be the associated Cartan decomposition. The global Cartan decomposition states that

K\times\mathfrak p\longrightarrow G,\qquad (k,X)\longmapsto k\exp X,

is a diffeomorphism. If $\mathfrak a\subseteq\mathfrak p$ is maximal abelian and $\overline{\mathfrak a^+}$ is a closed positive Weyl chamber, then every $g\in G$ can be written

g=k_1\exp(H)k_2,\qquad k_1,k_2\in K,\quad H\in\overline{\mathfrak a^+}.

This refinement is the $KAK$ decomposition.

Uniqueness

The polar coordinates $g=k\exp X$ are unique. In a $KAK$ expression the two compact factors need not be unique, but the chamber element $H$ is unique. Without restricting $H$ to a closed chamber, its orbit under the restricted Weyl group is the invariant datum Knapp, Chapters VI–VII.

Geometric meaning

The decomposition identifies $G/K$ with $\mathfrak p$ by $X\mapsto\exp(X)K$ , providing global normal coordinates on the associated Riemannian symmetric space. Passing from $\mathfrak p$ to a chamber in $\mathfrak a$ is the analogue of diagonalizing a positive-definite matrix. The resulting radial coordinate controls invariant integration formulas.

Example and applications

For $G=\mathrm{GL}_n(\mathbb R)$ and $K=\mathrm O(n)$ , the theorem is the matrix polar decomposition $g=k\exp X$ with $X$ symmetric. Orthogonally diagonalizing $X$ gives $KAK$ , with $A$ the positive diagonal matrices. In representation theory, the size of $H$ governs decay and growth estimates for matrix coefficients.

References

A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. Publisher record. Relevant: Chapter VI, §4 and Chapter VII, §8.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, 2001. DOI record. Relevant: Chapter IX on global decompositions.