Generalized affine Grassmannian slices, truncated shifted Yangians, and Hamiltonian reduction

Given a representation of a reductive group, 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.