The Mapping Class Group acts reducibly on SU(n)-character varieties

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 $L^2(Hom(\pi ,G)\mathrm{/}G,\mu )$ contains no finite-dimensional subrepresentations besides the constants. This note gives a short proof that when $G=SU(n)$, the representation $L^2(Hom(\pi ,G)\mathrm{/}G,\mu )$ contains many other invariant subspaces.