An overview of dualities in non-commutative harmonic analysis
Yulia Kuznetsova
TL;DR
This survey articulates how dualities extend Fourier-analytic methods beyond abelian groups, centering on the unitary dual $\\widehat{G}$ and its Plancherel measure to achieve $L^2$-isometries via $\\|f\\|_2^2 = \int_{\\widehat{G}} \mathrm{Tr}(\\widehat{f}(\\pi)^* \\widehat{f}(\\pi)) \, d\\mu(\\pi)$. It then develops symmetric-space analytics through the spherical transform, eigenfunction structure of invariant differential operators, and the Harish-Chandra c-function, while addressing non-unimodular and non-Type I subtleties with weights and disintegration. The discussion culminates in non-commutative $L_p$-theory and multiplier results, including $L^{r,\\infty}$-type criteria for $L^p-L^q$ boundedness and complete-boundedness criteria for group and quantum-group Fourier multipliers. Overall, the notes connect classical Fourier analysis with quantum-group dualities to outline a broad, practical panorama of tools for non-commutative harmonic analysis.
Abstract
These are edited notes of my mini-course given at the Analysis and PDE center of the University of Ghent, Belgium, in November 2024.