Limit multiplicities and von Neumann dimensions

Abstract

Given a connected semisimple Lie group $G$ and an arithmetic subgroup $\Gamma$, it is well known that each irreducible representation $\pi$ of $G$ occurs in the discrete spectrum $L_{\mathrm{disc}}^2(\Gamma \backslash G)$ of $L^2(\Gamma \backslash G)$ with at most a finite multiplicity $m_{\Gamma }(\pi )$. While $m_{\Gamma }(\pi )$ is unknown in general, we are interested in its limit as $\Gamma$ is taken to be in a tower of lattices ${\Gamma }_1\supset {\Gamma }_2\supset \cdots$. For a bounded measurable subset $X$ of the unitary dual $\hat{G}$, we let $m_{{\Gamma }_k}(X)$ be the integration of the multiplicity $m_{{\Gamma }_k}(\pi )$ over all $\pi$ in $X$, which can be proved finite. Let $H_X$ be the direct integral of the irreducible representations in $X$ with respect to the Plancherel measure of $\hat{G}$, which is also a module over the group von Neumann algebra $\mathcal{L}({\Gamma }_k)$. Based on the work of Sauvageot and Finis–Lapid–Müller, we prove

for any bounded subset $X$ of $\hat{G}$ when (i) $\{{\Gamma }_k{\}}_{k\ge 1}$ are cocompact or (ii) $G=\mathrm{SL}(n,\mathbb{R})$ and $\{{\Gamma }_k\}$ are principal congruence subgroups.