Hannes Wendt: A Duflo-Moore theorem for ergodic group actions on semifinite von Neumann algebras

Abstract: The orthogonality relations are a fundamental tool of representation theory. They have been extended by Duflo and Moore to the case of non-unimodular groups, and have found use also in branches of applied harmonic analysis, such as wavelet theory and time-frequency analysis.
Using the language of quantum harmonic analysis, their result can be stated as the integrability of certain operator convolutions. This viewpoint shows parallels to the properties of function convolutions of classical analysis. In recent joint work with Ulrik Enstad, we encompass both settings through ergodic, trace-preserving actions of locally compact groups on semifinite von Neumann algebras. I will present our main theorem for this general framework, which recovers the result of Duflo and Moore as well as Young's convolution inequality and the Fourier inversion formula as special cases.