Journal of Modern Dynamics (JMD)

Boundary unitary representations-irreducibility and rigidity

Pages: 49 - 69, Issue 1, January 2011      doi:10.3934/jmd.2011.5.49

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Uri Bader - Mathematics Department, The Technion - Israel Institute of Technology Haifa, 32000, Israel (email)
Roman Muchnik - Department of Mathematics and Computer Science, Lehman College, CUNY, 2500 Johnson Avenue Bronx, NY 10463, United States (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.
Mathematics Subject Classification:  Primary: 20C07, 22D40; Secondary: 37A30, 47A35.

Received: January 2010;      Revised: December 2010;      Available Online: April 2011.