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 $Sp(L,E)$, $σ_{δ,k}(L,E)$ and $σ_{π,k}(L,E)$. 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.

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