  When $G$ is a connected compact Lie group, and $\pi$ is a closed surface
group, then $Hom(\pi,G)$ contains an open dense $Out(\pi)$-invariant subset
which is a smooth symplectic manifold. This symplectic structure is
$Out(\pi)$-invariant and therefore defines an invariant measure $\mu$, which
has finite volume. The corresponding unitary representation of $Out(\pi)$ on
$L^2(Hom(\pi,G)/G,\mu)$ contains no finite-dimensional subrepresentations
besides the constants. This note gives a short proof that when $G=SU(n)$, the
representation $L^2(Hom(\pi,G)/G,\mu)$ contains many other invariant subspaces.
