$\mathrm{SU}(p,q)$ is know as "type AIII", see e.g. Goodman-Wallach, Helgason, or Knapp which may have the most details. For its action on projective space and other flag manifolds a classic reference is Wolf.

$\mathrm{SU}(p,q)$ is known as "type AIII", see e.g. Goodman-Wallach, Helgason, or Knapp which may have the most details. 

For its action on projective space and other flag manifolds a classic reference is Wolf. (If $pq\ne0$ then by Witt's theorem $\mathrm{SU}(p,q)$ has three orbits $P_+$, $P_-$, $P_0$ in projective space, consisting of the lines on which the defining hermitian form is positive, resp. negative, resp. zero. The former two are open while the latter one is closed.)