Twisting functors and Gelfand--Tsetlin modules over semisimple Lie algebras

Abstract

We associate to an arbitrary positive root \(\alpha\) of a complex semisimple finite-dimensional Lie algebra \(\mfrak{g}\) a twisting endofunctor \(T_\alpha\) of the category of \(\mfrak{g}\)-modules. We apply this functor to generalized Verma modules in the category \(\mcal{O}(\mfrak{g})\) 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 \(\mfrak{g}\) generated by a Cartan subalgebra of \(\mfrak{g}\) and by the Casimir element of the \(\mfrak{sl}(2)\)-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 \(\mfrak{g}\) is a complex simple Lie algebra and \(\alpha\) is the maximal root of \(\mfrak{g}\). 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 \(\mfrak{g}\) 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 \(\mcal{O}(\mfrak{g})\).