Two New Settings for Examples of von Neumann Dimension

Abstract

Let $G=PSL(2,\mathbb{R})$, let $\Gamma$ be a lattice in $G$, and let $\mathcal{H}$ be an irreducible unitary representation of $G$ with square-integrable matrix coefficients. A theorem in [Goodman, de la Harpe, Jones 1989] states that the von Neumann dimension of $\mathcal{H}$ as a $R\Gamma$-module is equal to the formal dimension of the discrete series representation $\mathcal{H}$ times the covolume of $\Gamma$, calculated with respect to the same Haar measure. We prove two results inspired by this theorem. First, we show there is a representation of $R\Gamma_2$ on a subspace of cuspidal automorphic functions in $L^2(\Gamma_1 \backslash G)$, where $\Gamma_1$ and $\Gamma_2$ are lattices in $G$; and this representation is unitarily equivalent to one of the representations in [Goodman, de la Harpe, Jones 1989]. Next, we calculate von Neumann dimensions when $G$ is $PGL(2,F)$, for $F$ a local non-archimedean field of characteristic $0$ with residue field of order not divisible by 2; $\Gamma$ is a torsion-free lattice in $PGL(2,F)$, which, by a theorem of Ihara, is a free group; and $\mathcal{H}$ is the Steinberg representation, or a depth-zero supercuspidal representation, each yielding a different dimension.