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On p.30 of the book "Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$".

Let $M_\mu:=\{v\in M: h\cdot v=\mu(h)v \ \forall h\in\mathfrak{h}\}$.

I quote: To make this precise, define a subspace of M for each fixed $\chi:Z(\mathfrak{g})\to\mathbb{C}$ by $M^\chi:= \{v \in M |\text{for each $z\in Z(\mathfrak{g})$}, (z - \chi(z))^n \cdot v = 0 \ \text{for some $n > 0$ depending on $z$}\}$.

In words, each element of $Z(\mathfrak{g})$ acts on $v$ as the scalar $\chi(z)$ plus a nilpotent operator. It is clear that $M^\chi$ is a $U(\mathfrak{g})$-submodule of $M$, while the subspaces $M^\chi$ are independent. Now $Z(\mathfrak{g})$ stabilizes each weight space $M_\mu$, since $Z(\mathfrak{g})$ and $U(\mathfrak{h})$ commute. Since we are working over an algebraically closed field, a standard result from linear algebra on families of commuting operators implies that $M_\mu=\bigoplus_{\chi}(M_\mu \cap M^\chi)$.

My question:

  1. What is the standard result from linear algebra on families of commuting operators meant?

  2. Does anyone have references on that standard result?

  3. It would be nice to have a sketch of proof for the direct sum decomposition $M_\mu=\bigoplus_{\chi}(M_\mu \cap M^\chi)$. Does anyone can tell me?

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  • $\begingroup$ Recall commuting operators preserves each other's (generalized) eigenspaces. So you look at intersecting generalized eigenspaces of different members of the family. I think this is done in his Intro to Lie Algebra book... $\endgroup$ Jun 4, 2019 at 15:45

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