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!
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!
I am not sure I get exactly what you mean, but the answer is probably no.
If $g$ is one-dimensional and we fix a non-zero element $X_0\in g$, then providing a $g$-module structure to a vector space $V$ is the same thing as picking an endomorphism $T\in\gl(V)$, corresponding to the action of $X_0$ on the module (i.e., what you call $\rho(X_0)$ is a comment above).
If you pick a linear map $T:V\to V$ which is not semisimple (that is, whose Jordan canonical form is not diagonal), then your «generalized eigenspaces of $V$» are surely the same thing as the generalized eigenspaces of $T$, and these are certainly not actual eigenspaces, because of the choice of $T$.