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Quaternionic Gamma functions and their logarithmic derivatives as spectral functions

Jean-Francois Burnol
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Abstract

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''.)

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Jean-Francois Burnol. 2000-11-10. Quaternionic Gamma functions and their logarithmic derivatives as spectral functions. https://arxiv.org/abs/math/9904044

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