Counting lattice points in norm balls on higher rank simple Lie groups (Q1651749)

Let \(n\geq 2\) and \(\tau: G=\mathrm{SL}_2(\mathbb R)\to \mathrm{GL}_n(\mathbb R)\) an irreducible representation. Then \(\tau\) indices a norm on \(G\), \[ \| x\|_\tau= \sqrt{\mathrm{tr}\tau(X)^t\tau(x)}. \] Let \(\Gamma\subset G\) be a lattice and write \(\kappa= \frac{1}{(n+1)(n+2)}\). For \(R>0\) let \(B^\tau_R\) be the set of all \(x\in G\) with \(\| x\|_\tau\leq R\). By \textit{W. Duke} et al. [Duke Math. J. 71, No. 1, 143--179 (1993; Zbl 0798.11024)] it is shown that for every \(\varepsilon>0\) one has \[ \frac{| B^\tau_R\cap\Gamma|}{\mathrm{vol}(B^\tau_R)}= \mathrm{vol}(\Gamma\setminus G)+O_\varepsilon\Biggl(\Biggl(\frac{1}{\mathrm{vol}(B^\tau_R)}\Biggr)^{\kappa-\varepsilon}\Biggr),\qquad R\to\infty. \] Using refined spectral estimates based on universal bounds for spherical functions, the authors of the current paper improve the error bound in various ways, in certain cases reaching an improvement of a factor 2, as the following example indicates: For \(\tau\) being the adjoint representation they show that \[ \frac{| B^{\mathrm{Ad}}_R\cap\Gamma|}{\mathrm{vol}(B^{\mathrm{Ad}}_R)}= \text{vol}(\Gamma\setminus G)+ O_\varepsilon \Biggl(\Biggl(\frac{1}{\text{vol}(B^\tau_R)}\Biggr)^{2\frac{n}{n+2}\kappa} \log(R)^q\Biggr),\quad R\to\infty \] for some explicit constant \(q\). Although they only present the case \(G=\mathrm{SL}_n\) the methods apply to any semisimple Lie group.