Iwasawa decomposition - Imported Knowlpedia Content

Statement

Let $G$ be a real reductive Lie group with maximal compact subgroup $K$ . Choose a maximal abelian subspace $\mathfrak a\subseteq\mathfrak p$ , positive restricted roots $\Sigma^+$ , and set

\mathfrak n=\bigoplus_{\alpha\in\Sigma^+}\mathfrak g_\alpha,\qquad A=\exp\mathfrak a,\qquad N=\exp\mathfrak n.

The Iwasawa decomposition is the theorem that multiplication

K\times A\times N\longrightarrow G,\qquad(k,a,n)\longmapsto kan,

is a diffeomorphism. In particular, every $g\in G$ has a unique factorization $g=k(g)a(g)n(g)$ relative to these choices.

Construction from restricted roots

The algebraic statement behind the theorem is $\mathfrak g=\mathfrak k\oplus\mathfrak a\oplus\mathfrak n$ as real vector spaces. Here $\mathfrak n$ is nilpotent because brackets add positive restricted roots. Exponentiation is a diffeomorphism from $\mathfrak n$ to the simply connected nilpotent group $N$ Knapp, Chapter VI, §5.

Minimal parabolic subgroup

Let $M=Z_K(A)$ . Then $P=MAN$ is a minimal parabolic subgroup, with $M$ compact modulo a possible finite central feature in the usual conventions. Characters and representations of $MAN$ , extended trivially across $N$ and induced to $G$ , produce principal-series representations. Thus $KAN$ is more than a coordinate system: it supplies the subgroup data used in parabolic induction.

Example and comparison

For $G=\mathrm{SL}_n(\mathbb R)$ , take $K=\mathrm{SO}(n)$ , $A$ the positive diagonal determinant-one matrices, and $N$ the upper unitriangular matrices. The decomposition becomes the QR factorization with a normalized diagonal. Unlike the $KAK$ decomposition, $KAN$ is uniquely ordered and asymmetric, reflecting the choice of positive roots.

References

A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. Publisher record. Relevant: Chapter VI, §5.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, 2001. DOI record. Relevant: Chapter IX on the Iwasawa decomposition.