Von Neumann Dimensions and Trace Formulas I: Limit Multiplicities

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 $L^2_{\text{disc}}(\Gamma\backslash G)$ 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 $\Gamma_1\supset \Gamma_2\supset\dots$. For a bounded measurable subset $X$ of the unitary dual $\widehat{G}$, we let $m_{\Gamma_n}(X)$ be the sum of the multiplicity $m_{\Gamma_n}(\pi)$ of a representation $\pi$ over all $\pi$ in $X$. Let $H_X$ be the direct integral of the irreducible representations in $X$, which is also a module over the group von Neumann algebra $\mathcal{L}\Gamma_n$. We prove: \begin{center} $\lim\limits_{n\to \infty}\cfrac{m_{\Gamma_n}(X)}{\dim_{\mathcal{L}\Gamma_n}H_X}=1$, \end{center} for any bounded subset $X$ of $\widehat{G}$, when i) $\Gamma_n$'s are cocompact, or, ii) $G=\SL(n,\mathbb{R})$ and $\{\Gamma_n\}$ are principal congruence subgroups.