Quaternionic Gamma Functions and Their Logarithmic Derivatives as Spectral Functions

We establish Connes’s local trace formula (related to the explicit formulae of number theory) for the quaternions. This is done as an application of a study of the central operator H = log(|x|) + log(|y|) in the context of invariant harmonic analysis. The multiplicative analysis of the additive Fourier transform gives a spectral interpretation to generalized “Tate Gamma functions” (closely akin to the Godement-Jacquet “γ(s, π, ψ)” functions.) The analysis of H leads furthermore to a spectral interpretation for the logarithmic derivatives of these Gamma functions (which are involved in “explicit formulae”.)