Question

I am looking for an embedding of the orthogonal Lie group O(n,C) into GL(m,C) such that the standard Iwasawa decomposition (also known as the QR-decomposition) for the group GL(m,C) induces an Iwasawa decomposition for the group O(n,C). Recall that the standard Iwasawa decomposition for the general linear group GL(m,C)=U(m)R, with R being the subgroup of upper-diagonal matrices with positive real diagonal entries, and U(m) - the unitary subgroup.

Answer 1

There are general theorems for an arbitrary semi-simple subgroup of $GL_n({\mathbb C})$ which do this. I will work this out for $O(n,{\mathbb C})$ when $n=2m$ is even.

Fix the standard inner product with respect to which the unitary group $U(n)$ is defined. View $O(n)=O(2m)$ as the subgroup which fixes the quadratic form

$$Q=x_1x_{2m}+x_2x_{2m-1}+\cdots+ x_mx_{m+1}.$$

All this is explained in Professor Jim Humphrey's book on Lie algebras.

Then the Iwasawa decomposition of $GL_n({\mathbb C})$ restriced to $O(n)$ gives an Iwasawa decomposition on $O(n)$.