  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 delta-functions" on the flag manifold G/B
supported at the 1-dimensional submanifold. When g=sl(n) (or gl(n)) 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).
