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

TL;DR

The paper proves that the algebra of asymptotic differential operators on the base affine space $SL_n/U$, $D_\hbar(SL_n/U)$, is isomorphic to the quantized Coulomb branch $\mathbb{C}_\hbar[\mathcal{M}_C(Q_n)]$ of a specific 3d $\mathcal{N}=4$ quiver theory, linking representation theory with quantum field theoretic constructions. In its semiclassical limit, this confirms a DHK21 conjecture about universal hyperkähler implosion for $SL_n$ and extends to a general family of unipotent reductions $T^*SL_n\sslash_\psi U$, identifying their Coulomb branches with the corresponding quiver theories. The work further clarifies the Gelfand–Graev action on $D_\hbar(SL_n/U)$ as arising from a permutation action on the Coulomb-branch side and develops a base-change framework for ring-objects in derived Satake categories to underpin these identifications. By leveraging results of GR15 and Mac23, the authors establish a robust interplay between geometric Satake, Braverman–Finkelberg–Nakajima Coulomb branches, and hyperkähler geometry, with potential extensions to other classical groups and connections to symplectic singularities.

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 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 $D_\hbar(SL_n/U)$.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Proposition 3.3
• proof : Proof of Proposition \ref{['prop:algiso']}
• Proposition 4.5
• Theorem 4.6
• Theorem 4.7
• proof : Proof of Theorem \ref{['thm:mainthmintro']}