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.


1 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 $O(n)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.