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

DOI10.1016/J.EXMATH.2022.09.001arXiv1901.11501MaRDI QIDQ6313370

Lauren C. Ruth

Publication date: 31 January 2019

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.

Mathematics Subject Classification ID

General theory of von Neumann algebras (46L10) Fuchsian groups and their generalizations (group-theoretic aspects) (20H10) Representations of Lie and linear algebraic groups over local fields (22E50)

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