# The operator algebra content of the Ramanujan–Petersson problem

### Florin Rădulescu

Università degli Studi di Roma Tor Vergata, Roma, Italy

## Abstract

Let $G$ be a discrete countable group, and let $Γ$ be an almost normal subgroup. In this paper we investigate the classification of (projective, with 2-cocycle $ε∈H_{2}(G,T)$) unitary representations $π$ of $G$ into the unitary group of the Hilbert space $l_{2}(Γ,ε)$ that extend the (projective, with 2-cocycle $ε$) unitary left regular representation of $Γ$. Representations with this property are obtained by restricting to $G$ (projective) unitary square integrable representations of a larger semisimple Lie group $Gˉ$, containing $G$ as dense subgroup and such that $Γ$ is a lattice in $Gˉ$. 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 $Γ$-invariant vectors for the unitary representation $π⊗πˉ$ is an intrinsic problem on the outer automorphism group of the skewed, crossed product von Neumann algebra $L(G⋊_{ε}L_{∞}(G,μ))$, where $G$ is the Schlichting completion of $G$ and $μ$ is the canonical Haar measure on $G$.

## Cite this article

Florin Rădulescu, The operator algebra content of the Ramanujan–Petersson problem. J. Noncommut. Geom. 13 (2019), no. 3, pp. 805–855

DOI 10.4171/JNCG/353