$\mathrm{SU}(p,q)$ is know as "type AIII", see e.g. [Goodman-Wallach](http://books.google.com/books?id=tbSX5VPE4PIC&pg=PA65), [Helgason](http://books.google.com/books?id=DWGvsa6bcuMC&pg=PA518), or [Knapp](http://books.google.com/books?id=U573NrppkA8C&pg=PA698) which may have the most details. For its action on projective space and other flag manifolds a classic reference is [Wolf](http://www.ams.org/mathscinet-getitem?mr=251246).