Differential Operators on the Base Affine Space of SLn and Quantized Coulomb Branches

Abstract We show that the algebra $D_\hbar (SL_{n}/U)$ of differential operators on the base affine space of $SL_{n}$ is the quantized Coulomb branch of a certain 3d $\mathcal{N} = 4$ quiver gauge theory. In the semiclassical limit this proves a conjecture of Dancer–Hanany–Kirwan about the universal hyperkähler implosion of $SL_{n}$. We also formulate and prove a generalization identifying the Hamiltonian reduction $T^{*} SL_{n}\ /\!/ _{\psi } U$ as a Coulomb branch for an arbitrary unipotent character $\psi $. As an application of our results, we provide a new interpretation of the Gelfand–Graev symmetric group action on $D_\hbar (SL_{n}/U)$.