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?



Equivalence of definition of category $\mathcal{O}^\mathfrak{p}$
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 $n \in U(\mathfrak{u})m.$ 


