Harmonic Analysis and Hecke Operators

Abstract. We first construct new uniform pointwise bounds for the matrix coeffi-cients of infinite dimensional unitary representations of a reductive algebraic group over a local field k with semisimple k-rank at least 2 [21]. We explain how this information on local harmonic analysis yields norm estimates for (global) Hecke operators on L2(Γ\G) for a connected almost simple simply connected Q-group G and its congruence subgroup Γ [5]. With the tool of Hecke operators, we settle a question of Linnik raised in the early sixties on the distribution of integer points of Diophantine type varieties when the varieties are homogeneous spaces of a reductive algebraic group over Q [10]. Lastly we discuss how to obtain evenly distributed se-quences on the spheres Sn (n ≥ 4) [22], generalizing the work of Lubotzky, Phillips and Sarnak on S2 and S3 ([18], [19]). 1 Uniform pointwise bounds ξS for matrix coefficients The aim of this section is to explain the construction of new uniform pointwise bounds for the matrix coefficients of infinite dimensional unitary representa-tions of a reductive algebraic group over a local field [21]. Let k be a local fiel