The limit multiplicities and von Neumann dimensions

Abstract: Given an arithmetic subgroup $\Gamma$ in a semi-simple Lie group $G$, the multiplicity of an irreducible representation of $G$ in $L^2(\Gamma\backslash G)$ is unknown in general. We observe the multiplicity of any discrete series representation $\pi$ of $\rm{SL}(2,\mathbb{R})$ in $L^2(\Gamma(n)\backslash \rm{SL}(2,\mathbb{R}))$ is close to the von Neumann dimension of $\pi$ over the group algebra of $\Gamma(n)$. We extend this result to other Lie groups and bounded families of irreducible representations of them. By applying the trace formulas, we show the multiplicities are exactly the von Neumann dimensions if we take certain towers of descending lattices in some Lie groups.

number theory

Audience: researchers in the topic

Harvard number theory seminar