Type II$_1$ von Neumann representations for Hecke operators on Maass forms and the Ramanujan-Peterson conjecture

Details Type II$_1$ von Neumann representations for Hecke operators on Maass forms and the Ramanujan-Peterson conjecture Type II$_1$ von Neumann representations for Hecke operators on Maass forms and the Ramanujan-Peterson conjecture

2008-02-25

arxiv.org/abs/0802.3548v12

Mathematics

Number Theory

To the original version two appendices have been added that explain the reduction step, which allows when computing essential

Scientific paper

Classical Hecke operators on Maass forms are unitarely equivalent, up to a commuting phase, to completely positive maps on II$_1$ factors, associated to a pair of isomorphic subfactors, and an intertwining unitary. This representation is obtained through a quantization procedure of the Hecke operators, which, in this representation, act on the Berezin's quantization deformation algebra of the fundamental domain of $\PSL(2,\Z)$ in the upper halfplane. The Hecke operators are inheriting from the ambient, non-commutative algebra on which they act, a rich structure of matrix inequalities. Using this construction we obtain that, for every prime $p$, the essential spectrum of the classical Hecke operator $T_p$ is contained in the interval $[-2\sqrt p, 2\sqrt p]$, predicted by the Ramanujan Petersson conjectures. In particular, given an open interval containing $[-2\sqrt p, 2\sqrt p]$, there are at most a finite number of possible exceptional eigenvalues lying outside this interval. The main tool for obtaining these representation of the Hecke operators (unitarely equivalent, up to commuting phase) is the square root of the state on $\PSL_2(\Q)$ measuring the displacement of fundamental domain. The square root is obtained from the matrix coefficients of the discrete series of $\PSL_2(\R)$ restricted to $\PSL_2(\Q)$ ([GH]).

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