Eskin, Alex; Oh, Hee Ergodic theoretic proof of equidistribution of Hecke points. (English) Zbl 1092.11023 Ergodic Theory Dyn. Syst. 26, No. 1, 163-167 (2006). 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 \(\Gamma\setminus G\). Reviewer: Florin Nicolae (Berlin) Cited in 1 ReviewCited in 22 Documents 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) Keywords:algebraic group; arithmetic subgroup; Hecke points × Cite Format Result Cite Review PDF Full Text: DOI arXiv