The Operator Algebra content of the Ramanujan-Petersson Problem

Abstract

Let $G$ be a discrete countable group, and let $\Gamma$ be an almost normal subgroup. In this paper we investigate the classification of (projective) unitary representations $\pi$ of $G$ into the unitary group of the Hilbert space $l^2(\Gamma)$ that extend the left regular representation of $\Gamma$. Representations with this property are obtained by restricting to $G$ square integrable representations of a larger semisimple Lie group $\bar G$, containing $G$ as dense subgroup and such that $\Gamma$ is a lattice in $\bar G$. This type of unitary representations of of $G$ appear in the study of automorphic forms. We prove that the Ramanujan-Petersson problem regarding the action of the Hecke algebra on the Hilbert space of $\Gamma$-invariant vectors for the unitary representation $\pi\otimes \bar\pi$ is an intrinsic problem on the outer automorphism group of the von Neumann algebra $\mathcal L(G \rtimes L^{\infty}(\mathcal G,\mu))$, where $\mathcal G$ is the Schlichting completion of $G$ and $\mu $ is the canonical Haar measure on $\mathcal G$.