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Let $g$ a Lie algebra and $V$ a finite-dimensional irreducible $g$-module, then each generalized eigenspace of $V$ is actually an eigenspace? If not, what is a condition to guarantee this fact?

Thanks!

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    Please do not cross-post: http://mathoverflow.net/questions/55871/eigenspaces-of-a-representation2011-02-18
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    Each generalized eigenspace of what?2011-02-18
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    I am sorry but that did not help: what is a "generalized eigenspace of $V$"?2011-02-18
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    Let $\rho: g \mapsto gl(V)$ such representation. The generalized eigenspace of $V$ via $\rho$ is $$V_i=\left\{v\in V \mid \forall x\in g, \exists n\in \mathbb N \text{ such that } (\rho(X) - \lambda_i(X))^nv = 0\right\}.$$2011-02-18
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    @user2764: add that to the body of the question, so that everybody can see it without having to read through all the comments. Presumably, you fixed a linear form $\lambda:g\to\mathbb C$ and are describing the "generalized eigenspace corresponding to $\lambda$"?2011-02-18

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