I have the following question:
Let $\mathfrak{g}:= \mathfrak{gl}(3)$ be the general linear algebra and $\mathfrak{gl}(2) \cong \mathfrak{a} \subset \mathfrak{gl}(3)$ a sub-algebra. Let $\mathcal{O}^{\mathfrak{g}}$ and $\mathcal{O}^{\mathfrak{a}}$ are categories $\mathcal{O}$ of them respectively. Is it true that $\mathcal{O}^{\mathfrak{g}} \subseteq \mathcal{O}^{\mathfrak{a} }$? More precisely, is it true that the restriction $\text{Res}^{\mathfrak{g}}_{\mathfrak{a}} (V) \in \mathcal{O}^{\mathfrak{a}} $ if provided that $V\in \mathcal{O}^{\mathfrak{g}}$? Thank you very much!
(Here the definition of the category $\mathcal{O}$ is given as following. Let $\mathcal{s}$ be a finite-dimensional, semi-simple Lie algebra over $\mathbb{C}$. Fix a Bore sub-algebra $\mathfrak{b}$ of $\mathfrak{g}$. Let $\mathfrak{h} \subset \mathfrak{b}$ be a Cartan sub-algebra of $\mathfrak{g}$ with the nilradical $\mathfrak{n}$. Then $W\in \mathcal{O}$ if and only if (1). $W$ is $\mathfrak{h}$-semi-simple. (2). $W$ is locally $\mathfrak{n}$-finite. (3). $W$ is finitely-generated (as $\mathfrak{g}$-module).)
Provided the answer to Tobias' question is yes, i.e. you take a standard embedding ${\mathfrak g}{\mathfrak l}(2)\subset{\mathfrak g}{\mathfrak l}(3)$, Properties (1) and (2) are preserved, but (3) is in general not: First, note that under the assumption of Properties (1) and (2), Property (3) is equivalent to $W$ being finitely generated over the ${\mathfrak n}_-$. Now, consider a Verma module $V$ over ${\mathfrak g}{\mathfrak l}(3)$: As a ${\mathscr U}{\mathfrak n}_-$-module, it is isomorphic to the regular module ${\mathscr U}({\mathfrak n}_-)$, which (by PBW, for example) is free of infinite rank over ${\mathscr U}{\mathfrak n}^{\prime}_-$, the universal enveloping algebra of the lower nilpotent part ${\mathfrak n}^{\prime}_-$ of ${\mathfrak g}{\mathfrak l}(2)$.