The operator algebra content of the Ramanujan–Petersson problem

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 epsilon is an element of H-2 (G,T)) unitary representations pi of G into the unitary group of the Hilbert space l(2) (Gamma, epsilon) that extend the (projective, with 2-cocycle epsilon) 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 (G) over bar, containing G as a dense subgroup and such that Gamma is a lattice in (G) over bar. This type of unitary representations 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 circle times (pi) over bar is an intrinsic problem on the outer automorphism group of the skewed, crossed product von Neumann algebra L(G (sic)(epsilon) L-infinity(g, mu)), where g is the Schlichting completion of G and mu is the canonical Haar measure on g.