This is a soft-question, but I haven't found an answer anywhere: do the factors of the Iwasawa decomposition of the pseudo-orthogonal group SO(p, q) have a simple form, in the same way that the factors of the Iwasawa decomposition of SL(n) do? I've only ever seen SL(n) given as an example, and I wondered why.



Iwasawa decomposition of the pseudo-orthogonal group
Indeed, it is easy to write out the Iwasawa decomposition for orthogonal groups $SO(p,q)$, say with $p\ge q$. By a change of variables, we can make the quadratic form $\pmatrix{ 0\_q & 0 & 1\_q \\ 0 & -1\_{p-q} & 0 \\ 1\_q & 0 & 0\_q}$. The maximal compact $K=S(O(p)\times O(q))$, which was diagonal blocks with the quadratic form $\pmatrix{-1\_p & 0 \\ 0 & 1\_q}$, gets mixed around in these coordinates, but now the standard minimal parabolic becomes visible: it is the collection of $\pmatrix{A & \* & \* \\ 0 & k & \* \\ 0 & 0 & A^{t-1}}$, where there are relations, e.g., $k\in SO(p-q)$. But/and the Levi component becomes apparent: $GL(n,\mathbb R)\times SO(p-q)$.


