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.
