The operator algebra content of the Ramanujan–Petersson problem
Florin Rădulescu
Università degli Studi di Roma Tor Vergata, Roma, Italy
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Abstract
Let be a discrete countable group, and let be an almost normal subgroup. In this paper we investigate the classification of (projective, with 2-cocycle ) unitary representations of into the unitary group of the Hilbert space that extend the (projective, with 2-cocycle ) unitary left regular representation of . Representations with this property are obtained by restricting to (projective) unitary square integrable representations of a larger semisimple Lie group , containing as dense subgroup and such that is a lattice in . This type of unitary representations of of 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 -invariant vectors for the unitary representation is an intrinsic problem on the outer automorphism group of the skewed, crossed product von Neumann algebra , where is the Schlichting completion of and is the canonical Haar measure on .
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