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arXiv · 1901.11501

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

Lauren C. Ruth
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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.

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BibTeXRIS

Lauren C. Ruth. 2019-01-31. The product of lattice covolume and discrete series formal dimension: p-adic GL(2). https://arxiv.org/abs/1901.11501

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