# About locally finite condition in category $\mathcal{O}^\mathfrak{p}$

Denote by $$\mathfrak{g}$$ a complex semisimple Lie algebra and let $$\mathfrak{h}$$ be a Cartan subalgebra of $$\mathfrak{g}$$.

Denote by $$\Phi$$ the root system of $$(\mathfrak{g},\mathfrak{h})$$ and denote by $$\mathfrak{g}_\alpha$$ the root subspace of $$\mathfrak{g}$$ corresponding to a root $$\alpha$$.

We fix a choice of the set of positive roots $$\Phi^+$$, and let $$\Delta$$ be the corresponding subset of simple roots in $$\Phi^+$$. Note that each subset $$I\subseteq\Delta$$ generates a root system $$\Phi_I\subseteq\Phi$$, with positive roots $$\Phi_I^+=\Phi_I\cap \Phi^+$$.

There are a number of subalgebras of $$\mathfrak{g}$$ associated with the root system $$\Phi_I$$. Let $$\mathfrak{l}_I=\mathfrak{h}\oplus\sum_{\alpha\in\Phi_I}\mathfrak{g}_\alpha$$ be the Levi subalgebra and let $$\mathfrak{u}_I=\sum_{\alpha\in\Phi^+\backslash\Phi_I^+}\mathfrak{g}_\alpha$$ be the nilpotent radical. Then $$\mathfrak{p}_I=\mathfrak{l}_I\oplus \mathfrak{u}_I$$ is the standard parabolic subalgebra.

We note that once $$I$$ is fixed, there is little use for other subsets of $$\Delta$$. Therefore, we omit the subscript if a subalgebra is obviously associated to $$I$$.

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.

I have search the notation: locally finite.

I get the following:

Locally finite modules $$M$$ are modules whose finitely generated submodules are finite (as sets).

I would like to know is that the correct definition for the locally finite in the definition of $$\mathcal{O}^\mathfrak{p}$$, secondly, I would like to know what is the intuitions/motivations for one to define $$\mathcal{O}^\mathfrak{p}$$ in such a way. Other two conditions seems more natural than the third one to me.

• For the definition of locally finite you should replace 'finite (as sets)' by 'finite dimensional (as vector spaces)' in your boldface definition. Equivalently every element of $M$ lives in a $U(\mathfrak{p})$-submodule that is finite dimensional as a vector space. – Simon Wadsley Dec 19 '18 at 12:08
• I think that, as often, the motivation comes from the examples. i.e. all three conditions hold in a large family of interesting examples of $U(\mathfrak{g})$-modules. – Simon Wadsley Dec 19 '18 at 12:11

3b. the nilradical of $$\mathfrak{p}$$ acts locally nilpotently on $$M$$