Twisting functors and Gelfand-Tsetlin modules over semisimple Lie algebras

Author(s)

Futorny, Vyacheslav ; Krizka, Libor

• Document type: Journal article
• Source: COMMUNICATIONS IN CONTEMPORARY MATHEMATICS; v. N/A, p. 37-pg., 2022-08-29.

Abstract

We associate to an arbitrary positive root a of alpha complex semisimple finite-dimensional Lie algebra g a twisting endofunctor T-alpha of the category of g-modules. We apply this functor to generalized Verma modules in the category O(g) and construct a family of alpha-Gelfand-Tsetlin modules with finite Gamma(alpha)-multiplicities, where Gamma(alpha) is a commutative C-subalgebra of the universal enveloping algebra of g generated by a Cartan subalgebra of g and by the Casimir element of the gl(2)-subalgebra corresponding to the root alpha. This covers classical results of Andersen and Stroppel when alpha is a simple root and previous results of the authors in the case when g is a complex simple Lie algebra and alpha is the maximal root of g. The significance of constructed modules is that they are Gelfand-Tsetlin modules with respect to any commutative C-subalgebra of the universal enveloping algebra of g containing Gamma(alpha). Using the Beilinson-Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of alpha-Gelfand-Tsetlin modules which contains constructed modules as well as the category O(g). (AU)

• FAPESP's process: 18/23690-6 - Structures, representations, and applications of algebraic systems
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants