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We show that the algebra $D_\hbar(SL_n/U)$ of differential operators on thebase 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 aconjecture of Dancer-Hanany-Kirwan about the universal hyperk\"ahler implosionof $SL_n$. We also formulate and prove a generalization identifying theHamiltonian reduction of $T^* SL_n$ with respect to an arbitrary unipotentcharacter as a Coulomb branch. As an application of our results, we provide anew interpretation of the Gelfand-Graev symmetric group action on$D_\hbar(SL_n/U)$.

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