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, with 2-cocycle \varepsilon\in H^2(G,\mathbb T) ) unitary representations \pi of G into the unitary group of the Hilbert space l^2(\Gamma, \varepsilon) that extend the (projective, with 2-cocycle \varepsilon ) unitary left regular representation of \Gamma . Representations with this property are obtained by restricting to G (projective) unitary 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 obtain a classification of such (projective) unitary representations and hence we obtain 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 skewed, crossed product von Neumann algebra {\mathcal L(G \rtimes_{\varepsilon} L^{\infty}(\mathcal G,\mu))} , where \mathcal G is the Schlichting completion of G and \mu is the canonical Haar measure on \mathcal G .