Gelfand-Tsetlin modules in the Coulomb context (Q6632634)

The author gives a new perspective on the theory of principal Galois orders. Every principal Galois order can be written as \(eFe\) for any idempotent \(e\) in an algebra \(F\), which is called a flag Galois order. In most important cases, one may 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 results about these algebras.\N\NThe author also discusses 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 \(U(\mathfrak{gl}_n)\). 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 one to interpret the classes of simple Gelfand-Tsetlin modules in terms of dual canonical bases for the Grothendieck group. In particular, the author classifies the Gelfand-Tsetlin modules over \(U(\mathfrak{gl}_n)\) and relate their characters to a generalization of Leclerc's shuffle expansion for dual canonical basis vectors. As an application, the author disproves a conjecture of Mazorchuk, showing that the fiber over a maximal ideal of the Gelfand-Tsetlin subalgebra appearing in a finite-dimensional representation has an infinite-dimensional module in its fiber for \(n \geq 6\).