Appears in collection : Bourbaki - Mars 2025

We shall discuss a recent work of Ben-Zvi, Sakellaridis, and Venkatesh which proposes a new paradigm for the relative Langlands program. The relative Langlands program is traditionally associated with the study of periods integrals of automorphic forms and their relation to analytic properties of $L$-functions. An earlier work of Sakellaridis and Venkatesh had proposed that the framework for this study should be that of spherical varieties. Ben-Zvi, Sakellaridis, and Venkatesh propose a larger framework for the relative Langlands program, that of hyperspherical varieties, which is a class of symplectic varieties with a Hamiltonian group action. With this larger framework, they envision a duality operation on hyperspherical varieties which explains many examples and phenomena already studied in the literature. This purported duality is partially motivated by a duality of boundary conditions induced by the $S$-duality of $4d$ topological quantum field theories, via its connection with the geometric Langlands duality.

[After Ben-Zvi, Sakellaridis, and Venkatesh]