In this paper we give a proof of the Hecke quantum unique ergodic-ity rate conjecture for the Berry-Hannay model. A model of quantum mechanics on the 2-dimensional torus. In the paper " Quantization of linear maps on the torus-Fresnel diffrac-tion by a periodic grating " , published in 1980 (see [BH]), the physicists M.V. Berry and J. Hannay explore a model for quantum mechanics on the 2-dimens...

We consider a family of Hecke C∗-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group. Suppose that M is a subgroup of a gro...

The main result of [Ash-Stevens 97] describes a framework for constructing padic analytic families of p-ordinary arithmetic Hecke eigenclasses in the cohomology of congruence subgroups of GL(n)/Q where the Hecke eigenvalues vary p-adic analytically as functions of the weight. Unanswered in that paper was the question of whether or not every p-ordinary arithmetic eigenclass can be “deformed” in ...

Let f be a normalized Hecke–Maass cusp form of weight zero for the group $$SL_2({\mathbb {Z}})$$ . This article presents several quantitative results about distribution Hecke eigenvalues f. Applications to $$\Omega _{\pm }$$ -results and its symmetric square sym $$^2(f)$$ are also given.

The determination of the Iwahori-spherical unitary representations for split p-adic groups can be reduced to the classification of unitary representations with real infinitesimal character for associated graded Hecke algebras. We determine the unitary modules with real infinitesimal character for the graded Hecke algebra of type E6.

Irreducible characters of Hecke algebras of type A may be represented as reened counts of simple statistics on suitable subsets of permutations. Such formulas have been generalized to characters of other Coxeter groups and their Hecke algebras and to coinvariant algebras. In this paper we present several formulas, applications to combinatorial identities, and related problems. New results are g...

In this short note we extend results of Kohnen and Sengupta on the sign of eigenvalues of Siegel cuspforms. We show that their bound for the first negative Hecke eigenvalue of a genus 2 Siegel cuspform of level 1 extends to the case of level N > 1. We also discuss the signs of Hecke eigenvalues of CAP forms.

In this paper we prove a version of Deligne’s conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.

We use the results on the minimal basis of the centre of an Iwahori-Hecke algebra from our earlier work, as well as some additional results on the minimal basis, to describe the image and kernel of the Brauer homomorphism defined by L. Jones for Iwahori-Hecke algebras of type A.