Langlands duality for character varieties via BPS cohomology

In this talk, I will introduce and explain a proof of the Langlands duality conjecture for local systems on the three-dimensional torus. The argument takes advantage of the local structure of the moduli stack of local systems, expressed in terms of stacks of commuting elements in a Lie algebra, together with the cohomological integrality isomorphism, to compute the BPS sheaf associated with the commuting variety. The computation has a Lie-theoretic flavor, involving the classification of distinguished nilpotent orbits in the Lie algebra (due to Premet). It also involves the classification of (quasi-)isolated elements in reductive groups (due to Bonnafé, Digne--Michel). Our study of the BPS sheaf also has an application to the topological mirror symmetry conjecture for G-Higgs bundles on elliptic curves. These conjectures were introduced in Tasuki Kinjo's talk, and this talk is based on joint work with Tasuki Kinjo.