Harmonic analysis on $p$-adic spherical varieties - Part 1/2
Appears in collection : 2025 - T1 - WS3 - Analysis on homogeneous spaces and operator algebras
The first lecture will present a general formalism for theorems in harmonic analysis, which applies not only to spherical varieties but also to other spaces of polynomial growth (such as finite-volume quotients of semisimple Lie groups). We will discuss the variation of discrete series, asymptotic cones (boundary degenerations), and how to build the $L^2$ and Harish-Chandra Schwartz spaces out of those ingredients.
The second lecture will focus on arithmetic aspects of harmonic analysis. We will discuss how (local) $L$-functions show up in scattering operators and Plancherel densities, and conjectures about the parametrization of the spectrum by means of the "dual group" of a spherical variety.
