The Taylor spectrum of Lie algbera representations

In 1970, J. L. Taylor introduced a joint spectrum for a commuting $n$-tuple of operators acting on a Banach space. What makes the Taylor spectrum interesting is that there is a holomorphic functional calculus defined in any neighborhood of the spectrum and satisfying the spectral mapping property. We generalize the Taylor spectrum to operator families generating a finite-dimensional Lie algebra $\mathfrak g$, and we analyze the spectrum in the case where the operators act on a finite-dimensional space. In particular, we completely describe the spectrum in the cases where $\mathfrak g$ is either nilpotent or semisimple, and also for some "nice" representations of the Borel subalgebra of a semisimple Lie algebra.