Geometric construction of Gelfand--Tsetlin modules over simple Lie algebras

TL;DR

This paper introduces a new geometric approach to constructing Gelfand--Tsetlin modules for complex simple Lie algebras, providing explicit realizations and criteria for simplicity, especially for sl(3,C).

Contribution

It presents a novel geometric construction of Gelfand--Tsetlin modules applicable to all complex simple Lie algebras, extending previous algebraic methods.

Findings

Constructed Gelfand--Tsetlin modules as delta-functions on flag manifolds.

Provided simplicity criteria for modules over sl(3,C).

Extended the class of known Gelfand--Tsetlin modules to infinite-dimensional cases.

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 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).