## 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.