Iwasawa decomposition


In mathematics, the Iwasawa decomposition 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. It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.

Definition

Then the Iwasawa decomposition of is
and the Iwasawa decomposition of G is
meaning there is an analytic diffeomorphism from the manifold to the Lie group, sending.
The dimension of A is equal to the real rank of G.
Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a maximal compact subgroup provided the center of G is finite.
The restricted root space decomposition is
where is the centralizer of in and is the root space. The number
is called the multiplicity of.

Examples

If G=SLn, 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 symplectic group G=Sp, 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 : In this case, the group can be written as a product of the subgroup of upper-triangular matrices and the subgroup, where is the ring of integers of.