Relative Langlands duality - Part 1/2
Appears in collection : 2025 - T1 - WS3 - Analysis on homogeneous spaces and operator algebras
In these two lectures I will give an overview of the relative Langlands duality conjectures, as they are formulated in my work with Yiannis Sakellaridis and Akshay Venkatesh. The rough idea is that multiplicity-free harmonic analysis problems (spherical varieties and their variants) come in pairs associated to Langlands dual groups, so that representation theoretic questions on one side are matched with questions of spectral geometry on the other. This matching is expected to be realized in each of the settings of the Langlands program (global, local, arithmetic and geometric). The classical setting of harmonic analysis over local fields (``local, arithmetic'') is thus sandwiched between the global arithmetic setting, which pairs period integrals with special values of L-functions, and the local geometric setting, which provides instructions for building the dual geometry out of a sheaf-theoretic form of harmonic analysis.
