# Equivalence of definition of category $\mathcal{O}^\mathfrak{p}$

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

Let $$\Phi$$ be the root system of $$(\mathfrak{g},\mathfrak{h})$$, write $$W$$ for the corresponding Weyl group of $$\Phi$$, and denote by $$\mathfrak{g}_\alpha$$ the root subspace of $$\mathfrak{g}$$ corresponding to a root $$\alpha$$.

We fix a choice 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 a standard parabolic subalgebra of $$\mathfrak{g}$$.

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 $$I$$.

Definition 1: The category $$\mathcal{O}^\mathfrak{p}$$ is the full subcategory of $$U(\mathfrak{g})$$-Mod whose objects $$M$$ satisfy 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.

Definition 2: The category $$\mathcal{O}^\mathfrak{p}$$ is the full subcategory of $$U(\mathfrak{g})$$-Mod whose objects $$M$$ satisfy 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 $$U(\mathfrak{u})$$-module.

My question: How to show Definition 1 is equivalent to Definition 2?

By the PBW theorem we can write $$U(\mathfrak{p}) = U(\mathfrak{l})U(\mathfrak{u}).$$ By our assumption $$U(\mathfrak{u})m$$ is finite dimensional for any $$m\in M$$ and by the second point of the definition $$U(\mathfrak{l})n$$ is finite-dimensional for any $$n \in U(\mathfrak{u})m.$$