Differential operators on the base affine space of $SL_n$ and quantized Coulomb branches

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 hyperk\"ahler implosion of $SL_n$. We also formulate and prove a generalization identifying the Hamiltonian reduction of $T^* SL_n$ with respect to an arbitrary unipotent character as a Coulomb branch. As an application of our results, we provide a new interpretation of the Gelfand-Graev symmetric group action on $D_\hbar(SL_n/U)$.