In arXiv:2108.03453 with Ivan Losev and Lucas Mason-Brown we gave a definition of a unipotent representation of a complex semisimple Lie group G using quantizations of affinizations of nilpotent covers. The next goal is to understand the structure of the spherical unitary dual of G as something centered around the defined unipotent representations. Roughly, we want to describe (from geometric perspective) the “complementary series” around a unipotent representation.
The results of arXiv:2108.03453 give the description of unipotent representations completely in terms of the Lie group G, while most results in the literature are given in terms of the Langlands dual Lie algebra. To relate our work we need to provide a dictionary between the two descriptions. A part of this goal was achieved in arXiv:2309.14853. However, this project is far from complete, and continues to be a subject of my work.
We expect that certain varieties obtained by intersecting a Slodowy slice to one orbit with the cover of a different orbit are symplectic dual to each other. A special case of this situation suggests a symplectic duality between Slodowy slices and affinizations of covers of special orbits, mentioned in arXiv:2108.03453. Then the equivariant Hikita conjecture explains how special unipotent central characters naturally arise from the symplectic duality point of view. The work on Hikita conjecture was partially carried out in our paper arXiv:2410.20512 with Vasily Krylov and Do Kien Hoang. The general case is not understood yet.
In a joint project with Shilin Yu we aim to interpret Sommers and Achar duality maps via symplectic duality by considering extra symmetries on slices to symplectic leaves. This construction suggests existence of an order relation on the set of admissible covers of symplectic leaves in a conical symplectic singularity, and an order reversing bijection between these sets for symplectic dual varieties, such that the corresponding Hasse diagrams are dual to each other. We focus on the case of nilpotent cones of Langlands dual groups, that are expected to be symplectic dual.
The proof of Hikita conjecture for ADE quiver gauge theories suggests a deeper relation between equivariant cohomology of Nakajima quiver variety and B-algebra of quantized Coulomb branch. Namely, both are identified with weight spaces of dual Weil modules of current Lie algebra. Similarly, Proudfoot conjecture in this setting is a part of geometric Satake. It is interesting to see if other known instances of Hikita and Proudfoot conjectures fit in a similar setting. Currently, I am thinking about the version of Hikita conjecture for affine Grassmanian slices for non-simply-laced groups through representations of their current algebras.
The decomposition theorem proved by Ivan Losev gives an alternative approach to studying finite W algebras. The key idea is a generalization of he Darboux-Weinstein theorem. Roughly, if we have a quantization of a Poisson variety, then locally in a (formal) neighborhood of a point it decomposes into a product of a quantization of the symplectic leaf (which around the point looks just like a symplectic vector space), and a quantization of the slice to it. Applying it to a semisimple Lie algebra g with a natural Poisson structure in a beighborhood of a nilpotent element , we get a passage between universal enveloping algebra of g to the finite W algebra, associated with the nilpotent. In ongoing work with Ivan Losev we extend this construction to get a connection between any two nilpotent orbits, proving what is called "inverse Hamiltonian reduction" in literature in all cases. In ongoing work with Zachary Carlini and Ilya Dumanski we extend all these constructions to the setting of affine W algebras.