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 $Sp(L,E)$, $\sigma_{\delta,k}(L,E)$ and $\sigma_{\pi,k}(L,E)$. 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.