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

Abstract

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.