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

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 $\mcal{O}(\mfrak{g})$.