Boundary unitary representations - irreducibility and rigidity

Abstract

Let $M$ be compact negatively curved manifold, $\Gamma ={\pi }_1(M)$ and $\tilde{M}$ be its universal cover. Denote by $B=\partial \tilde{M}$ the geodesic boundary of $\tilde{M}$ and by $\nu$ the Patterson-Sullivan measure on $X$. In this note we prove that the associated unitary representation of $\Gamma$ on $L^2(B,\nu )$ is irreducible. We also establish a new rigidity phenomenon: we show that some of the geometry of $M$, namely its marked length spectrum, is reflected in this $L^2$-representations.