Gelfand-Tsetlin theory for rational Galois algebras
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GELFAND–TSETLIN THEORY FOR RATIONAL GALOIS ALGEBRAS
BY
Vyacheslav Futorny Instituto de Matem´ atica e Estat´ıstica, Universidade de S˜ ao Paulo S˜ ao Paulo SP, Brasil e-mail: [email protected]
AND
Dimitar Grantcharov Department of Mathematics, University of Texas at Arlington Arlington, TX 76019, USA e-mail: [email protected]
AND
Luis Enrique Ramirez Universidade Federal do ABC, Santo Andr´e-SP, Brasil e-mail: [email protected]
AND
Pablo Zadunaisky Universidad CAECE, Buenos Aires, Argentina e-mail: [email protected]
Received June 1, 2019 and in revised form July 29, 2019
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V. FUTORNY ET AL.
Isr. J. Math.
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 gl(n) and the finite W -algebras of type A.
1. Introduction The category of Gelfand–Tsetlin modules of the general linear Lie algebra gl(n, C) is an important category of modules that plays a prominent role in many areas of mathematics and theoretical physics. By definition, a Gelfand– Tsetlin module of gl(n) is one that has a generalized eigenspace decomposition over a certain maximal commutative subalgebra (Gelfand–Tsetlin subalgebra) Γ of the universal enveloping algebra of gl(n). This algebraic definition has a nice combinatorial flavor. The concept of a Gelfand–Tsetlin module generalizes the classical realization of the simple finite-dimensional representations of gl(n) via the so-called Gelfand–Tsetlin tableaux introduced in [GT50]. The explicit nature of the Gelfand–Tsetlin formulas inevitably raises the question of what infinite-dimensional modules admit tableaux bases—a question that led to the systematic study of the theory of Gelfand–Tsetlin modules. This theory has attracted considerable attention in the last 30 years of the 20th century and has been studied in [DOF91, DFO94, Maz98, Maz01, Mol99, Zhe73], among others. Gelfand–Tsetlin bases and modules are also related to Gelfand–Tsetlin integrable systems that were first introduced for the unitary Lie algebra u(n) by Guillemin and Sternberg in [GS83], and later for the general linear Lie algebra gl(n) by Kostant and Wallach in [KW06a] and [KW06b].
Vol. TBD, 2020
GELFAND–TSETLIN THEORY
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Recently, the study of Gelfand–Tsetlin modules took a new direction after the theory of singular Gelfand–Tsetlin modules was initiated in [FGR16]. Singular Gelfand–Tsetlin modules are roughly those that have basis of tableaux whose entries may be zeros of the denominators in the Gelfand–Tsetlin formulas. For the last three years remarkable
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