Higgs and Coulomb branches: Geometry and Representation Theory

Higgs and Coulomb branches of quiver gauge theories form two important families of Poisson varieties that are expected to be exchanged under so-called 3D mirror symmetry. Quantized Coulomb branches are associative algebras deforming the algebras of functions on Coulomb branches. They are closely related to many important representation-theoretic structures, such as Yangians, quantum groups, and Hecke algebras. In this talk, I will discuss how 3D mirror symmetry, together with other insights motivated by physics, yields very explicit answers to purely representation-theoretic questions about representations of some of these quantum groups. Talk is based on joint works with Dinkins, Karpov, Klyuev, and Lance.