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 $eFe$ for any idempotent $e$ in an algebra $F$, 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 results 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 $U({\mathfrak{gl}}_n)$. 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 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. Finally, as an application, we disprove 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 $n\ge 6$.

49 pages; v5. We found an error in the "proof" of the conjecture mentioned in the abstract, and now provide a counter-example disproving it. This is the final version, to appear in Annals of Representation Theory

References in corpus (3)

• Principal Galois orders and Gelfand-Zeitlin modules
• Generators for Coulomb branches of quiver gauge theories
• Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems

Cited by in corpus (6)

• Coulomb branches of quiver gauge theories with symmetrizers
• Generators for Coulomb branches of quiver gauge theories
• Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems
• Gelfand-Tsetlin modules: canonicity and calculations
• An extension of $U({\mathfrak{gl}}_n)$ related to the alternating group and Galois orders
• Gelfand-Tsetlin modules for gl(n,m)