Von Neumann Dimensions and Trace Formulas I: Limit Multiplicities
Given a connected semisimple Lie group $G$ and an arithmetic subgroup $Γ$, it is well-known that each irreducible representation $π$ of $G$ occurs in the discrete spectrum $L^2_{\text{disc}}(Γ\backslash G)$ of $L^2(Γ\backslash G)$ with at most a finite multiplicity $m_Γ(π)$. While $m_Γ(π)$ is unknown in general, we are interested in its limit as $Γ$ is taken to be in a tower of lattices $Γ_1\supset Γ_2\supset\dots$. For a bounded measurable subset $X$ of the unitary dual $\widehat{G}$, we let $m_{Γ_n}(X)$ be the sum of the multiplicity $m_{Γ_n}(π)$ of a representation $π$ over all $π$ 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 $\mathcal{L}Γ_n$. We prove: \begin{center} $\lim\limits_{n\to \infty}\cfrac{m_{Γ_n}(X)}{\dim_{\mathcal{L}Γ_n}H_X}=1$, \end{center} for any bounded subset $X$ of $\widehat{G}$, when i) $Γ_n$'s are cocompact, or, ii) $G=\SL(n,\mathbb{R})$ and $\{Γ_n\}$ are principal congruence subgroups.