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.
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$\begingroup$ This is discussed in Helgason's book on symmetric spaces. $\endgroup$– VenkataramanaMar 3, 2014 at 1:41
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$\begingroup$ I can not find anywhere in this book which answers my questions! $\endgroup$– user47700Mar 3, 2014 at 2:10
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$\begingroup$ One can choose a Cartan to consist of diagonal imaginary matrices with trace $0$, then roots correspond to matrices with two nonzero entries that are off the diagonal in opposite positions that are conjugate or anti-conjugate to each other. You should be able to figure everything out from that. $\endgroup$– Will SawinMar 3, 2014 at 2:30
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$\mathrm{SU}(p,q)$ 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.)
answered Mar 3, 2014 at 2:37