Von Neumann Dimensions and Trace Formulas I: Limit Multiplicities

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 $L_{\text{disc}}^2(\Gamma \backslash G)$ 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 ${\Gamma }_1\supset {\Gamma }_2\supset \dots$. For a bounded measurable subset $X$ of the unitary dual $\hat{G}$, 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} $\underset{n\to \infty }{\lim }\frac{m_{{\Gamma }_n}(X)}{{\dim }_{\mathcal{L}{\Gamma }_n}{H}_X}=1$, \end{center} for any bounded subset $X$ of $\hat{G}$, when i) ${\Gamma }_n$'s are cocompact, or, ii) $G=\SL(n,\mathbb{R})$ and $\{{\Gamma }_n\}$ are principal congruence subgroups.

15 pages