The Support of an Irreducible Lie Algebra Representation

We present an explicit description of the -support suppMof any irreducible -locally finite -moduleM, where is any finite-dimensional Lie algebra and is an arbitrary nilpotent Lie subalgebra of . If contains a Cartan subalgebra of the semi-simple part of , we reformulate the description of suppMin terms of a lattice Mand of the convex hullSMof suppM. When is reductive it is known that suppMis nothing but the intersection ofSMwith the root lattice Δ shifted by an arbitrary element of suppM. Our general description is similar, but the root lattice Δ must be replaced by a certain sublattice M, and suppMmay now have “holes” near the boundary ofSM. The paper is concluded by a detailed discussion of examples.