Questions about the geometry of G-representation varieties on a manifold M have attracted many researchers as the theory combines the algebraic geometry of G, the topology of M, and the group theory and representation theory of G and the fundamental group of M. In this talk, I will explain how to construct a Topological Quantum Field Theory to compute virtual classes of character stacks (G-representation varieties equipped with the adjoint G-action) in the Grothendieck ring of stacks. I will also show a few features of the construction (for instance, how to obtain arithmetic information) focusing on a couple of simple examples.
The work is joint with Jesse Vogel and Ángel González-Prieto.  

Symplectic duality predicts that affine symplectic singularities come in pairs that are in a sense dual to each other. The Hikita conjecture relates the cohomology of the symplectic resolution on one side to the functions on the fixed points on the dual side.  

In a recent work with Ivan Losev and Lucas Mason-Brown, we suggested an important example of symplectic dual pairs. Namely, a Slodowy slice to a nilpotent orbit should be dual to an affinization of a certain cover of a special orbit for the Langlands dual group. In that paper, we explain that the appearance of the special unipotent central character can be seen as a manifestation of a slight generalization of the Hikita conjecture for this pair.

However, a further study shows that several things can (and do!) go wrong with the conjecture. In this talk, I will explain a modified version of the statement, recent progress towards the proof, and how special unipotent characters appear in the picture. It is based on a work in progress with Do Kien Hoang and Vasily Krylov.