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Ergodic theoretic proof of equidistribution of Hecke points. (English) Zbl 1092.11023

Let \(G\) be a connected non-compact \(\mathbb Q\)-simple real algebraic group defined over \(\mathbb Q\), \(\Gamma\subset G(\mathbb Q)\) an arithmetic subgroup of \(G\), \(\text{Comm}(\Gamma)\) the commensurator group of \(\Gamma\). For an element \(a\in\text{Comm}(\Gamma)\) the \(\Gamma\)-orbit \(\Gamma\setminus\Gamma a \Gamma\) in \(\Gamma\setminus G\) has finitely many points called Hecke points associated with \(a\). Seting \(\text{deg}(a)=\#\Gamma\setminus\Gamma a \Gamma,\) it holds \(\text{deg}(a)=[\Gamma:\Gamma\cap a^{-1}\Gamma a].\) The authors prove: Let \(a_i\in \text{Comm}(\Gamma)\) be a sequence such that \(\text{lim}_{i\rightarrow\infty}\text{deg}(a_i)= \infty.\) Then for any bounded continuous function \(f\) on \(\Gamma\setminus G\) and for any \(x\in \Gamma\setminus G\), \[ \text{lim}_{i\rightarrow\infty} \frac{1}{\text{deg}(a_i)}\sum_{ \gamma\in\Gamma\setminus\Gamma a_i \Gamma}f(\gamma x)=\int_{\Gamma\setminus G}f(g)\,d\mu_G(g), \] where \(\mu_G\) is the \(G\)-invariant Borel probability measure on \(\Gamma\setminus G\).

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E40 Discrete subgroups of Lie groups
37A15 General groups of measure-preserving transformations and dynamical systems
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)