Representing some II${}_1$ factors in $L^2(\Lambda \backslash G)$

Abstract

Let $G$ be $PGL(n,F)$, $n\ge 3$, $F$ a certain non-archimedean local field; or let $G$ be $PSL(2,\mathbb{R})\times \cdots \times PSL(2,\mathbb{R})$. Let $\Gamma$ be a lattice in $G$, and let $({\Lambda }_n)$ be a sequence of lattices in $G$ satisfying the pointwise limit multiplicity property. In this note, we explain how the pointwise limit multiplicity property can be combined with a generalization of a theorem in \cite{ghj} to give representations of the II${}_1$ factor $R\Gamma$ on a subspace of $L^2({\Lambda }_i\backslash G)$ for some ${\Lambda }_i$ in $({\Lambda }_n)$. This extends a result in the author's dissertation \cite{ruthphd}.

5 pages