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

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?

• You haven't defined $\Phi_I^+$ or $\mathcal{O}^{\mathfrak{p}}$. Do you want us to look in the book you cite? – S. Carnahan Mar 7 at 15:14
• Sorry for forgetting to define. – James Cheung Mar 9 at 9:47

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 $$\mathcal{O}^{\mathfrak{p}}.$$

• Thank you very much. By the way, is locally $n_I^-$-finite meant any finitely generated $n_I^-$-module is finite dimensional as a vector space? – James Cheung Mar 9 at 9:52
• Yes, you can phrase it like that. – Vít Tuček Mar 10 at 12:05
• @JamesCheung When the Verma module is not in the parabolic subcategory, one often considers their images under the so-called Zuckerman functors as suitable replacements (I was going to point you to my paper with Mazorchuk on parabolic projective functors, but I can see that we hardly give any details about this there, so probably there are better references, likely written by Mazorchuk and collaborators). – Tobias Kildetoft Mar 31 at 19:27
• @TobiasKildetoft I would be interested in that if you ever come across it. – Vít Tuček Mar 31 at 20:22
• After taking another look, it seems that Mazorchuk's "Homological Properties ... II" should provide at least a starting point for some citation hopping, but it does seem like it will take some steps to get to any details on the actual functors (other than just the definition). – Tobias Kildetoft Mar 31 at 20:42

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.

• Thank you for spotting my mistake and for writing such a useful book! – Vít Tuček Mar 29 at 9:24