# A standard result from linear algebra on families of commuting operators

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?

• 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... Jun 4, 2019 at 15:45