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 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(glnβ). 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(glnβ) 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