Hyperfunctions and Harmonic Analysis on Symmetric Spaces

“…Being closely related to that of [10,Lemma 4.6], it is based on a characterization of ίρ in terms of finite dimensional representations as in [14]. Our treatment follows the presentation in [26].…”

Section: Erik P Van Den Banmentioning

confidence: 98%

A convexity theorem for semisimple symmetric spaces

Ban¹

1986

Pacific J. Math.

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“…More generally, an element g ∈ G is called δ-regular if its A-component is δ-regular. Note that the A + -component of an element is uniquely defined, and the K-and H-components of a regular element are uniquely defined modulo the subgroup M which is the centraliser of A in K ∩ H. We refer to [Sc,Ch. 7] and [HS, Part II] for basic facts about affine symmetric spaces.…”

Section: Lattice Points On Affine Symmetric Varietiesmentioning

confidence: 99%

Counting lattice points

Gorodnik

Nevo

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

104

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Abstract. For a locally compact second countable group G and a lattice subgroup Γ, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and the domains satisfy a basic regularity condition, ii) the mean ergodic theorem for the action of G on G/Γ holds, with a rate of convergence.The error term we establish matches the best current result for balls in symmetric spaces of simple higher-rank Lie groups, but holds in much greater generality.A significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap. In particular, the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property τ .We discuss a number of applications, including: counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties.We note that the mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze-Stein phenomenon. We formulate and prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary step in our proof of quantitative results in counting rational points.

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Hyperfunctions and Harmonic Analysis on Symmetric Spaces

Cited by 96 publications

References 0 publications

Mixing, counting, and equidistribution in Lie groups

Mixing, counting, and equidistribution in Lie groups

A convexity theorem for semisimple symmetric spaces

Counting lattice points

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