# The product of lattice covolume and discrete series formal dimension: p-adic GL(2)

## TL;DR

This paper computes the product of lattice covolume and formal dimension for discrete series representations of p-adic GL(2), linking it to von Neumann dimensions via cohomological and group-theoretic methods.

## Contribution

It provides a method to explicitly calculate the product of covolume and formal dimension for discrete series in p-adic GL(2), connecting geometric and representation-theoretic invariants.

## Key findings

- The product equals the von Neumann dimension of the representation.
- The covolume is derived from Ihara's theorem.
- Formal dimensions are based on results by Corwin, Moy, and Sally.

## Abstract

Let $F$ be a nonarchimedean local field of characteristic $0$ and residue field of order not divisible by $2$. We show how to calculate the product of the covolume of a torsion-free lattice in $PGL(2,F)$ and the formal dimension of a discrete series representation of $GL(2,F)$. The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.