Questions to the proof of Lemma 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 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.

Answer 1

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 $U(\mathfrak{n}_I^-).v$.