  For a compact Riemann surface $X$ of genus $g > 1$, $\Hom(\pi_1(X),
PU(p,q))/PU(p,q)$ is the moduli space of flat $PU(p,q)$-connections on $X$.
There are two invariants, the Chern class $c$ and the Toledo invariant $\tau$
associated with each element in the moduli. The Toledo invariant is bounded in
the range $-2min(p,q)(g-1) \le \tau \le 2min(p,q)(g-1)$. This paper shows that
the component, associated with a fixed $\tau > 2(max(p,q)-1)(g-1)$ (resp. $\tau
< -2(max(p,q)-1)(g-1)$) and a fixed Chern class $c$, is connected (The
restriction on $\tau$ implies $p=q$).
