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.
