  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
$D_\hbar(SL_n/U)$.
