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Let $F$ be a nonarchimedean local field of characteristic $0$ and residuefield of order not divisible by $2$. We show how to calculate the product ofthe covolume of a torsion-free lattice in $PGL(2,F)$ and the formal dimensionof a discrete series representation of $GL(2,F)$. The covolume comes from atheorem 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 thesecond cohomology group of a free group, the resulting product is the vonNeumann dimension of a discrete series representation considered as arepresentation of a free group factor.

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