Verma modules in category $\mathcal{O}^\mathfrak{p}$

Question

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Fix a Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$ and fix a parabolic subalgebra $\mathfrak{p}$ containing $\mathfrak{b}$. Let $I \subseteq\Delta$ be the subset of simple roots corresponding to $\mathfrak{p}$. Denote by $\Phi_I$ the subsystem generated by $I$. i.e., $\Phi_I=\Phi\cap\sum_{\alpha\in I}\mathbb{Z}\alpha$. Let $\Phi^+_I=\Phi_I\cap\Phi^+$.

Let $\mathfrak{l} = \mathfrak{h}\oplus\sum_{\alpha\in \Phi_I}\mathfrak{g}_\alpha$ be the Levi subalgebra. Denote by $\mathfrak{u}$ the nilpotent radical of $\mathfrak{p}$ and let $\overline{\mathfrak{u}}$ be the dual space of $\mathfrak{u}$. Note that $\mathfrak{p}=\mathfrak{l}\oplus \mathfrak{u}$.

The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple $\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$.

The category $\mathcal{O}^\mathfrak{p}$ is the full subcategory of $U(\mathfrak{g})$-Mod such that every object $M$ in category $\mathcal{O}^\mathfrak{p}$ satisfies the following conditions.

1. $M$ is a finitely generated $U(\mathfrak{g})$-module.
2. $M$ is a direct sum of finite-dimensional simple $U(\mathfrak{l})$-modules.
3. $M$ is locally finite as a $U(\mathfrak{p})$-module.

The Verma module is of the form $M(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda$, where $\mathbb{C}_\lambda$ is a simple $\mathfrak{b}$-module with weight $\lambda$. Denote by $L(\lambda)$ the unique simple quotient of $M(\lambda)$.

The parabolic Verma module is defined to be $M_I(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{p})} F(\lambda)$, where $F(\lambda)$ is the simple finite-dimensional $\mathfrak{l}$-module with highest weight $\lambda$.

The set of $\Phi^+_I$-dominant integral weights in $\mathfrak{h}^*$ is defined to be $\Lambda^+_I = \{\lambda \in \mathfrak{h}^* : \langle\lambda,\alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+_I\}$.

By Proposition 9.3 and Theorem 9.4 in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, we get $L(\lambda)\in\mathcal{O}^\mathfrak{p}\iff \lambda\in\Lambda_I^+$. Also $\lambda\in \Lambda_I^+ \implies M_I(\lambda)\in \mathcal{O}^\mathfrak{p}$.

What about $M(\lambda)$? Does $\lambda\in \Lambda_I^+$ imply $M(\lambda)\in \mathcal{O}^\mathfrak{p}$? If not, any counterexample?

Answer 1

According to Lemma 9.3, a module $M$ is in $\mathcal{O}^{\mathfrak{p}}$ if and only if it is locally $\mathfrak{n}_I^-$-finite. Since the Verma module $M(\lambda)$ contains infinite-dimensional $\mathfrak{U}(\mathfrak{n}_I^-)$-submodule $\mathfrak{U}(\mathfrak{n}_I^-)\otimes \mathbb{C}_\lambda$, we see that it is never in $\mathcal{O}^{\mathfrak{p}}.$

Answer 2

Vit's answer is mostly correct but overlooks the extreme case when $I$ is empty: then $\mathcal{O}^{\mathfrak{p}} = \mathcal{O}$. (At the other extreme one gets all finite dimensional modules.) It's alwsys worth checking these extreme cases, though they often don't say much about the general case.

I do apologize for the fragmentary state of Chapter 9 (and the variation of notation and termnilogy in the scattered literature, which I'm partly responskble for). In fact there are many unanswered questions, and it isn't always clear how important these questions are. But saying nothing at all didn't seem to be an option at the time. There is more literature now, but it's still rather scattered and incomplete.