TITLE: Root space decomposition
QUESTION [5 upvotes]: What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form 
$\left(
  \begin{array}{cc}
    X & Y \\
    \overline{Y}^t & Z \\
  \end{array}
\right)
$, where $\overline{X}^t=-X$, $\overline{Z}^t=-Z$, and $tr(X)+tr(Z)=0$. Also is there any reference regarding the action of the Lie group $SU(p, q)$ on complex projective space? Thanks.

REPLY [7 votes]: $\mathrm{SU}(p,q)$ 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.)