Gelfand-Tsetlin theory for rational Galois algebras

Abstract

In the present paper we study Gelfand-Tsetlin modules defined in terms of BGG differential operators. The structure of these modules is described with the aid of the Postnikov-Stanley polynomials introduced in [PS09]. These polynomials are used to identify the action of the Gelfand-Tsetlin subalgebra on the BGG operators. We also provide explicit bases of the corresponding Gelfand-Tsetlin modules and prove a simplicity criterion for these modules. The results hold for modules defined over standard Galois orders of type A—a large class of rings that include the universal enveloping algebra of \(\mathfrak{gl}\)(n) and the finite W-algebras of type A.