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Mathematics, Representation Theory
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 $A$.
