Questions to the proof of Proposition 9.3 in Humphreys “Representations of Semisimple Lie algebras in the BGG Category $\mathcal{O}$"

Question

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ with Cartan subalgebra $\mathfrak{h}$, root system $\Phi \subset \mathfrak{h}^*$ and Weyl group $W$. Fix a set of positive roots $\Phi^+ \subset \Phi$ and simple roots $\Delta \subset \Phi$. This defines a Cartan decomposition $\mathfrak{g}=\mathfrak{n}^- \oplus \mathfrak{h}\oplus \mathfrak{n} $ with $\mathfrak{n}^- =\bigoplus_{\alpha \in -\Phi^+}$ and $\mathfrak{n} =\bigoplus_{\alpha \in \Phi^+}\mathfrak{g}_\alpha $. Furthermore let $\mathfrak{b}=\mathfrak{h}\oplus \mathfrak{n}$. Then we have for $I \subset \Delta$ a standard parabolic subalgebra $\mathfrak{p}_I \supset \mathfrak{b}$. Moreover it defines a root system $\Phi_I \subset \Phi$ with positive roots $\Phi_I^+ \subset \Phi^+$ and a Weyl group $W_I \subset W$.

Additionally we have the following subalgebras of $\mathfrak{g}$:

• $\mathfrak{l}_I=\mathfrak{h}\oplus \sum_{\alpha \in \Phi_I}\mathfrak{g}_\alpha$,
• $\mathfrak{u}_I=\bigoplus_{\alpha \in \Phi^+\backslash \Phi_I^+}\mathfrak{g}_\alpha$,
• $\mathfrak{g}_I=[\mathfrak{l}_I,\mathfrak{l}_I]$,
• $\mathfrak{h}_I=\bigoplus_{\alpha \in I} \mathbb{C}h_\alpha$,
• $\mathfrak{n}_I=\bigoplus_{\alpha \in \Phi_I^+}\mathfrak{g}_\alpha$,
• $\mathfrak{n}_I^-=\bigoplus_{\alpha \in -\Phi_I^+}\mathfrak{g}_\alpha$

such that

• $\mathfrak{p}_I=\mathfrak{l}_I \oplus \mathfrak{u}_I$ and
• $\mathfrak{g}_I=\mathfrak{n}_I^- \oplus \mathfrak{h}_I \oplus \mathfrak{n}_I$.

In Humphreys "Representations of Semisimple Lie algebras in the BGG Category $\mathcal{O}$" we have the following Lemma 9.3:

Let $M \in \mathcal{O}$ have the set of weights $\Pi(M)$. The following conditions are equivalent:

1. $M$ is locally $\mathfrak{n}^-$-finite.
2. For all $\alpha \in I$ and $\mu \in \Pi(M)$ we have $\dim M_\mu=\dim M_{s_{\alpha}\mu}$.
3. For all $w \in W_I$ and $\mu \in \Pi(M)$ we have $\dim M_\mu=\dim M_{w\mu}$.
4. $\Pi(M)$ is stable under $W_I$.

to which already asked a question and gave a more detailed proof (Questions to the proof of Lemma 9.3 in Humphreys "Representations of Semisimple Lie algebras in the BGG Category $\mathcal{O}$").

Now I try to understand the following part of Proposition 9.3:

$M\in \mathcal{O}$ lies in $\mathcal{O}^\mathfrak{p}$ if and only if $M$ satisfies one of the equivalent conditions of Lemma 9.3.

I think it is not necessary to explain what $\mathcal{O}^\mathfrak{p}$ is as my question breaks down to the following claim:

• $M$ being locally $n_I^-$-finite implies that $M$ is a direct sum of finite dimensional simple $U(\mathfrak{l}_I)$-modules.

Humphreys proof goes as follows. By assumption any weight vector $v \in M$ generates a finite-dimensional $U(\mathfrak{g}_I)$-modules $N$ on which $\mathfrak{h}$ acts semisimply since $\mathfrak{h}$ normalizes $\mathfrak{n}_I$ and $\mathfrak{n}_I^-$. Then it follows that every element of $M$ lies in a finite dimensional $U(\mathfrak{l}_I)$-module $N'$. By complete reducibility, a "standard" argument shows that $M$ is a direct sum of simple $U(\mathfrak{l}_I)$-modules.

Question that arises for me:

1. Why acts $\mathfrak{h}$ semisimply on $N$? Or rather why is it enough that $\mathfrak{h}$ normalizes $\mathfrak{n}_I$ and $\mathfrak{n}_I^-$?
2. Why does $\mathfrak{h}$-semisimplicity of $N$ implies then that every element of $M$ lies in a finite dimensional $U(\mathfrak{l}_I)$-module?
3. What is this standard argument that implies that $M$ is a direct sum of simple $U(\mathfrak{l}_I)$-modules?