Given a representation of a reductive group, Braverman-Finkelberg-Nakajima have defined a remarkable Poisson variety called the Coulomb branch. Their construction of this space was motivated by considerations from supersymmetric gauge theories and symplectic duality. The coordinate ring of this Coulomb branch is defined as a kind of cohomological Hall algebra; thus it makes sense to develop a type of “Springer theory” to define modules over this algebra. In this talk, we will explain this BFN Springer theory and give many examples. In the toric case, we will see a beautiful combinatorics of polytopes. In the quiver case, we will see connections to the representations of quivers over power series rings. In the general case, we will explore the relations between this Springer theory and quasimap spaces.[-]

Given a representation V of a reductive group G, Braverman-Finkelberg-Nakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This variety comes with a natural deformation quantization, called a Coulomb branch algebra. Important cases of these Coulomb branches are (generalized) affine Grassmannian slices, and their quantizations are truncated shifted Yangians.
Motivated by the geometric Satake correspondence and the theory of symplectic duality/3d mirror symmetry, we expect a categorical g-action on modules for these truncated shifted Yangians. I will explain three results in this direction. First, we have an indirect realization of this action, using equivalences with KLRW-modules. Second, we have a geometric relation between these generalized slices by Hamiltonian reduction. Finally, we have an algebraic version of this Hamiltonian reduction which we are able to relate to the first realization.[-]