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]).
Partially sponsored by NSF grant DMS 0070544
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Oh, H. (2002). Harmonic Analysis and Hecke Operators. In: Burger, M., Iozzi, A. (eds) Rigidity in Dynamics and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04743-9_19
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