Iwasawa decomposition

In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.
Definition.
Then the Iwasawa decomposition of formula_1 is
and the Iwasawa decomposition of "G" is
meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold formula_17 to the Lie group formula_18, sending formula_19.
The dimension of "A" (or equivalently of formula_6) is equal to the real rank of "G".
Iwasawa decompositions also hold for some disconnected semisimple groups "G", where "K" becomes a (disconnected) maximal compact subgroup provided the center of "G" is finite.
The restricted root space decomposition is
where formula_22 is the centralizer of formula_23 in formula_24 and formula_25 is the root space. The number
formula_26 is called the multiplicity of formula_27.
Examples.
If "G"="SLn"(R), then we can take "K" to be the orthogonal matrices, "A" to be the positive diagonal matrices with determinant 1, and "N" to be the unipotent group consisting of upper triangular matrices with 1s on the diagonal.
For the case of "n"="2", Iwasawa decomposition of "G"="SL(2,R)" is in terms of
For the symplectic group "G"="Sp(2n", R ")", a possible Iwasawa-decomposition is in terms of
Non-Archimedean Iwasawa decomposition.
There is an analog to the above Iwasawa decomposition for a non-Archimedean field formula_34: In this case, the group formula_35 can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup formula_36, where formula_37 is the ring of integers of formula_34.