UC Riverside

Let G=PSL(2, R), let Γ be a lattice in G, and let H be an irreducible unitary representation of G with square-integrable matrix coefficients. A theorem in [GHJ89] states that the von Neumann dimension of H as a RΓ-module is equal to the formal dimension of the discrete series representation H times the covolume of Γ, calculated with respect to the same Haar measure. We prove two results inspired by this theorem. First, we show there is a representation of RΓ2 on a subspace of cuspidal automorphic functions in L2(Γ1\G), where Γ1 and Γ2 are lattices in G; and this representation is unitarily equivalent to one of the representations in [GHJ89]. Next, we calculate von Neumann dimensions when G is PGL(2, F), for F a non-archimedean local field of characteristic 0 with residue field of order not divisible by 2; Γ is a torsion-free lattice in PGL(2, F), which, by a theorem of Ihara, is a free group; and H is the Steinberg representation, or a depth-zero supercuspidal representation, each yielding a different dimension.