##### Department of Mathematics,

University of California San Diego

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### Operator Algebra Seminar

## Lauren Ruth

#### UC Riverside

## 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 $W^*(\Gamma)$-module is equal to the formal dimension of the discrete series representation $\mathcal{H}$ times the covolume of $\Gamma$, both calculated with respect to the same Haar measure. We will present two results which take inspiration from this theorem. In the first part of the talk, we will show that there is a representation of $W^*(\Gamma)$ on a subspace of cuspidal automorphic functions in $L^2(\Lambda \backslash G)$, where $\Lambda$ is any other lattice in $G$ (and $W^*(\Gamma)$ acts on the right); and this representation is unitarily equivalent to one of the representations in Goodman--de la Harpe--Jones. In the second part of the talk, we will explain how the proof of the theorem in Goodman--de la Harpe--Jones carries over to a wider class of groups, including the situation where $G$ is $PGL(2,F)$, for $F$ a local non-archimedean field of characteristic $0$, and $\Gamma$ is a torsion-free lattice in $PGL(2,F)$, which, by a theorem of Ihara, is a free group. To give a simple example, we will focus on the case when $\mathcal{H}$ is the Steinberg representation (as opposed to a supercuspidal representation), and we will calculate its von Neumann dimension as a $W^*(\Gamma)$-module. This yields representations of free group factors that are not unitarily equivalent to those representations obtained in the setting of $PSL(2,\mathbb{R})$.

Host: Hans Wenzl

### August 30, 2017

### 11:00 AM

### AP&M 5829

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