GAAG seminar - Examples of relative Langlands duality for D-modules in positive characteristic

Using the Azumaya property of (crystalline) differential operators in positive characteristic, Bezrukavnikov-Braverman and Chen-Zhu established a generic version of the geometric Langlands correspondence, which is realized as a twisted version of the Fourier-Mukai transform on dual Hitchin systems. In the recent work of Ben-Zvi--Sakellaridis--Venkatesh, they proposed the relative Langlands duality: Given a Hamiltonian G-space and its dual, they constructed the so-called period sheaf and L-sheaf that conjecturally correspond with each other under the geometric Langlands correspondence. In the setting of D-modules in positive characteristic, I will discuss work in progress to establish a generic version of this correspondence for Hamiltonian G-spaces that arise from quasi-split real groups.