Geometric construction of Gelfand–Tsetlin modules over simple Lie algebras

Abstract

In the present paper we describe a new class of Gelfand–Tsetlin modules for an arbitrary complex simple finite-dimensional Lie algebra g and give their geometric realization as the space of ‘δ-functions’ on the flag manifold G/B supported at the 1-dimensional submanifold. When g=sl(n,C) (or gl(n,C)) these modules form a subclass of Gelfand–Tsetlin modules with infinite-dimensional weight subspaces. We discuss their properties and describe the simplicity criterion for these modules in the case of the Lie algebra sl(3,C).