Boundary unitary representations-irreducibility and rigidity
Uri Bader - Mathematics Department, The Technion - Israel Institute of Technology Haifa, 32000, Israel (email)
Abstract: Let $M$ be compact negatively curved manifold, $\Gamma =\pi_1(M)$ and $M$ be its universal cover. Denote by $B =\partial M$ the geodesic boundary of $M$ 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.
Keywords: Unitary representations, Negative curvature, Boundary representations,
Marked length spectrum.
Received: January 2010; Revised: December 2010; Published: April 2011.