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 $\epsilon \in H^2(G,\mathbb{T})$) 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 $\bar{G}$, 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 $\mathcal{L}(G{⋊}_{\epsilon }{L}^{\infty }(\mathcal{G},\mu ))$, where $\mathcal{G}$ is the Schlichting completion of $G$ and $\mu$ is the canonical Haar measure on $\mathcal{G}$.