Boundary unitary representations - irreducibility and rigidity

TL;DR

This paper proves the irreducibility of boundary unitary representations for compact negatively curved manifolds and reveals how the manifold's marked length spectrum influences these representations, indicating a deep link between geometry and representation theory.

Contribution

It establishes the irreducibility of boundary unitary representations and demonstrates a new rigidity phenomenon connecting the manifold's geometry to these representations.

Findings

The boundary representation on L^2(B,ν) is irreducible.

The marked length spectrum of the manifold influences the associated L^2-representations.

A new rigidity phenomenon links geometric data to representation properties.

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.