2019/02/06 by Futorny, Vyacheslav, Krizka, Libor
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1902.02269
We associate to an arbitrary positive root α of a complex semisimple finite-dimensional Lie algebra \mfrakg a twisting endofunctor Tα of the category of \mfrakg-modules. We apply this functor to generalized Verma modules in the category \mcalO(\mfrakg) and construct a family of α-Gelfand--Tsetlin modules with finite Γα-multiplicities, where Γα is a commutative \C-subalgebra of the universal enveloping algebra of \mfrakg generated by a Cartan subalgebra of \mfrakg and by the Casimir element of the \mfraksl(2)-subalgebra corresponding to the root α. This covers classical results of Andersen and Stroppel when α is a simple root and previous results of the authors in the case when \mfrakg is a complex simple Lie algebra and α is the maximal root of \mfrakg. 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 \mfrakg containing Γα. 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 α-Gelfand--Tsetlin modules which contains constructed modules as well as the category \mcalO(\mfrakg).