Joint spectra and nilpotent lie algebras of linear transformations

Given a complex nilpotent finite dimensional Lie algebra of linear transformations, L, in a complex finite dimensional vector space, E, we study the joint spectra Sp(L, E), σδ, k(L, E). and σπ, k(L, E). We compute them, and we prove that they all coincide with the set of weights of L for E. We also give a new interpretation of some basic module operations of the Lie algebra L in terms of the joint spectra.