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:


1. $M$ is locally $\mathfrak{n}\_I^-$-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 $W\_I$.


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:


1. Does he consider $N$ as a finite dimensional $\mathfrak{sl} (2,\mathbb{C})$-module?
2. Why is $N$ $\mathfrak{h}$-stable and why is this necessary?
3. 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 $\mathfrak{n}\_I^-$-finite.


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



Questions to the proof of Lemma 9.3 in Humphreys "Representations of Semisimple Lie algebras in the BGG Category $\mathcal{O}$"
Thanks to the outstanding help of LSpice I present a version of more detailed proof of the two parts above. Do not hesitate to point out mistakes.


"$(1) \Rightarrow (2)$": Fix $\alpha \in I$ and $\mu \in \Pi(M)$. Observe that for $\mu(h\_\alpha)=0$, we have $s\_\alpha\mu=\mu-\langle \mu, \alpha^{\vee}\rangle \alpha = \mu - \mu(h\_\alpha)\alpha=\mu$ and $(2)$ follows trivially. Hence we can assume $\mu(h\_\alpha) \neq 0$.
Then by assumption and as $M\in \mathcal{O}$ the action on $M\_\mu$ of the subalgebra $\mathfrak{s}\_\alpha \cong \mathfrak{sl}(2,\mathbb{C})$, generated by $x\_\alpha \in \mathfrak{g}\_\alpha$, $y\_\alpha \in \mathfrak{g}\_{-\alpha}$, produces a finite dimensional $U(\mathfrak{sl}(2,\mathbb{C}))$-submodule $N \subset M$.


For $v \in M\_\mu$, we have that $N \ni h\_\alpha.v=\mu(h\_\alpha)v$. Hence $M\_\mu \subset N$. For $x\_\alpha,y\_\alpha$ exists $n\_\alpha \in \mathbb{N}$ such that $x\_\alpha^{n\_\alpha+1}.N=y\_\alpha^{n\_\alpha+1}.N=0.$ Define
\begin{align\*}
\exp(x\_\alpha)&:=\sum\_{k=0}^{n\_\alpha} x\_\alpha^k/k! \in U(\mathfrak{sl},(2,\mathbb{C}))\\
\exp(y\_\alpha)&:=\sum\_{k=0}^{n\_\alpha} y\_\alpha^k/k! \in U(\mathfrak{sl}(2,\mathbb{C})),\\
s&:=\exp(x\_\alpha)\exp(-y\_\alpha)\exp(x\_\alpha) \in U(\mathfrak{sl}(2,\mathbb{C}))
\end{align\*}
Then as in the proof of Theorem 21.2 of Humphreys "Lie algebras and Representations Theory", we additionally have that $M\_{s\_\alpha\mu}=s.M\_\mu \subset N$.


As $N$ is finite dimensional, $N=\bigoplus N\_i$ with $N\_i$ simple $U(\mathfrak{sl}(2,\mathbb{C}))$-module. Observe that
\begin{align\*}
h.(x\_\alpha.v)&=(x\_\alpha h+\alpha(h)x\_\alpha).v=(\mu(h)+\alpha(h))x\_\alpha.v, \\
h.(y\_\alpha.v)&=(x\_\alpha h-\alpha(h)x\_\alpha).v=(\mu(h)-\alpha(h))y\_\alpha.v,
\end{align\*}
for $v \in M\_\mu, h \in \mathfrak{h}$. Thus $N$ is $\mathfrak{h}$-stable and $M\_\mu=\bigoplus (N\_i)\_\mu$ resp. $M\_{s\_\alpha\mu}=\bigoplus (N\_i)\_{s\_\alpha\mu}$ follows. But for simple $U(\mathfrak{sl}(2,\mathbb{C}))$-module $N\_i$ we know that $\dim((N\_i)\_{s\_\alpha\mu})=\dim((N\_i)\_\mu)$ and the claim follows.


"$(4) \Rightarrow (1)$": We want to show that $U(\mathfrak{n}\_I^-).v$ is finite dimensional for every $v \in M$. As $M$ is $\mathfrak{h}$-semisimple, we can assume that $v \in M\_\mu$ for some $\mu \in \Pi(M)$. Then by assumption $w\_I\mu \in \Pi(M)$ and only finitely many $\mathbb{Z}^+$-linear combinations of $\Phi\_I^+$ can be added to get a weight of $M$.Thus, as $\mu=w\_I(w\_I\mu)$ and $w\_I$ interchange $\Phi\_I^+$ and $\Phi\_I^-$, only finitely many $\mathbb{Z}^+$-linear combinations of $\Phi\_I^-$ can be added to $\mu$ to lie in $\Pi(M)$. Hence $U(\mathfrak{n}\_I^-)M\_\mu$ is finite dimensional, especially $U(\mathfrak{n}\_I^-).v$.


