ISM Discovery School: Langlands Correspondence for Spherical Varieties

October 4-7, 2023

The relative Langlands program was born out of the methods used to study automorphic L-functions by representing them by various integrals of automorphic forms. The work of Jacquet, D. Prasad, and others, highlighted the connections between Langlands functoriality and the problem of distinction - or, harmonic analysis on certain (almost) homogeneous G-spaces X, such as spherical varieties. The conjectures of Gan-Gross-Prasad and Ichino-Ikeda, based on work of Waldspurger and many others, revealed a pattern that relates global integrals of automorphic forms to local harmonic analysis. The work of Gaitsgory-Nadler and Sakellaridis-Venkatesh allowed the formulation of a general program, based on the dual group of a spherical variety.

The goal of this workshop will be to introduce the more recent work of Ben-Zvi-Sakellaridis-Venkatesh, which takes a step further, introducing a categorical version of the relative Langlands program. In it, the “period integrals” of automorphic forms are viewed as “global quantizations” of a Hamiltonian G-space M, and there is a “dual” Hamiltonian (\check{G})-space (\check{M}) whose quantization corresponds to an L-function. This is in line with ideas of Kapustin-Witten and, especially, Gaiotto, seeking a correspondence of “boundary conditions” in the quantum-field-theory-interpretation of the geometric Langlands program. The workshop will mostly focus on the analogous local conjecture: We will discuss “relative Satake transforms” for spherical varieties, and the relation of associated Plancherel densities to local L- functions, and formulate a categorical version of this relation. This is modelled on the derived geometric Satake equivalence due to Ginzburg-Drinfeld-Bezrukavnikov-Finkelberg, which we view as a categorification of Macdonald’s formula for zonal spherical functions and the unramified Plancherel measure.

The deadline to apply to participate is July 31, 2023.