Harmonic Analysis and Hecke Operators

Chapter

Abstract

We first construct new uniform pointwise bounds for the matrix coefficients of infinite dimensional unitary representations of a reductive algebraic group defined over a local field k with semisimple k-rank at least 2, [22]. We explain how this information on local harmonic analysis yields norm estimates for (global) Hecke operators on L 2(Γ\G) for a connected almost simple simply connected ℚ-group G and its congruence subgroups Γ, [6]. 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, [11]. Lastly we discuss how to obtain evenly distributed sequences on the spheres S n (n ≥ 4) [23], generalizing the work of Lubotzky, Phillips and Sarnak on S 2 and S 3 ([19], [20]).

Keywords

Convolution 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Hee Oh
    • 1
  1. 1.Mathematics departmentPrinceton UniversityPrincetonUSA

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