Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ with root system $\Phi$, Weyl group $W$ and Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha $. Fix a set of positive roots $\Phi^+ \subset \Phi$ and simple roots $\Delta \subset \Phi$. Then $I \subset \Delta$, defines a root system $\Phi_I \subset \Phi$ with positive roots $\Phi_I^+ \subset \Phi^+$ and a Weyl group $W_I \subset W$. Furthermore be $\mathfrak{n}_I^-=\bigoplus_{\alpha \in -\Phi_I^+}\mathfrak{g}_\alpha$.

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:

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

Now I struggle with the main parts of the proof.

In "$1. \Rightarrow 2.$" he argues the following way. For $\alpha \in I$ fixed, consider the action of the subalgebra generated by $x_\alpha \in \mathfrak{g}_\alpha$ and $y_\alpha \in \mathfrak{g}_{-\alpha}$, which is isomorphic to $\mathfrak{sl} (2,\mathbb{C})$, on $M_\mu$. This gives a finite dimensional submodule $N$ of $M$, which is stable under $\mathfrak{h}$. Then the standard theory for finite dimensional representations of $\mathfrak{sl} (2,\mathbb{C})$ yields $2.$

Questions that arise for me:

- Does he consider $N$ as a finite dimensional $\mathfrak{sl} (2,\mathbb{C})$-module?
- Why is $N$ $\mathfrak{h}$-stable and why is this necessary?
- I know how the simple $\mathfrak{sl} (2,\mathbb{C})$-modules look like but how does this imply 2. ?

In "$4. \Rightarrow 1.$" He constructed a weight $\mu'(=w_I\mu)$, such that only finitely many $\mathbb{Z}^+$-linear combinations of $-\Phi_I^+$ can be added to $\mu'$ to lie in $\Pi(M)$. Then he says that $\mu'$ is a typical weight of $M$, forcing $M$ to be locally $\mathfrak{n}^-$-finite.

- How does this weight forces $M$ to be locally $\mathfrak{n}^-$-finite? Have the impression there is kind of standard argument used.