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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 $$D_\hbar (SL_{n}/U)$$. 

Abstract We study two classes of morphisms in infinite type: tamely presented morphisms and morphisms with coherent pullback. These are generalizations of finitely presented morphisms and morphisms of finite Tor-dimension, respectively. The class of tamely presented schemes and stacks is restricted enough to retain the key features of finite-type schemes from the point of view of coherent sheaf theory, but wide enough to encompass many infinite-type examples of interest in geometric representation theory. The condition that a diagonal has coherent pullback is a natural generalization of smoothness to the tamely presented setting, and we show such objects retain many good cohomological properties of smooth varieties. Our results are motivated by the study of convolution products in the double affine Hecke category and related categories in the theory of Coulomb branches. 

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with $$\mathbf{g}$$ -vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type $$A_{1}^{(1)}$$ , we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.