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Gelfand-Tsetlin modules in the Coulomb context

Abstract

This paper gives a new perspective on the theory of principal Galois orders as developed by Futorny, Ovsienko, Hartwig and others. Every principal Galois order can be written as eFeeFe for any idempotent ee in an algebra FF, which we call a flag Galois order; and in most important cases we can assume that these algebras are Morita equivalent. These algebras have the property that the completed algebra controlling the fiber over a maximal ideal has the same form as a subalgebra in a skew group ring, which gives a new perspective to a number of result about these algebras. We also discuss how this approach relates to the study of Coulomb branches in the sense of Braverman-Finkelberg-Nakajima, which are particularly beautiful examples of principal Galois orders. These include most of the interesting examples of principal Galois orders, such as U(gln)U(\mathfrak{gl}_n). In this case, all the objects discussed have a geometric interpretation which endows the category of Gelfand-Tsetlin modules with a graded lift and allows us to interpret the classes of simple Gelfand-Tsetlin modules in terms of dual canonical bases for the Grothendieck group. In particular, we classify Gelfand-Tsetlin modules over U(gln)U(\mathfrak{gl}_n) and relate their characters to a generalization of Leclerc's shuffle expansion for dual canonical basis vectors. Finally, as an application, we confirm a conjecture of Mazorchuk, showing that the weights of the Gelfand-Tsetlin integrable system which appear in finite-dimensional modules never appear in an infinite-dimensional simple module.Comment: 37 pages; v3: minor improvements before submissio

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