# Jacquet module for Lie algebras?

Let $G$ be a reductive group and let $P$ be a parabolic subgroup with Levi $L$ and unipotent radical $N$. Let $F$ be a finite or local non-Archimedean field. The Jacquet Functor is a functor from $G(F)$-modules to $L(F)$-modules defined by $V \mapsto V_N:=V/<n.v-v>$, where $<n.v-v>$ denotes the vector subspace of $V$ generated by expression of the form $n.v-v$, where $n\in N(F)$ and $v\in V$. The Jacquet functor satisfies many nice properties: it is exact, it is adjoint to parabolic induction, etc. It plays a fundamental role in representation theory of finite and p-adic groups.

Now suppose $\mathfrak{g}$ is a reductive Lie algebra over $\mathbb{C}$, and let $\mathfrak{p}$ be a parabolic with Levi $\mathfrak{l}$ and nilpotent radical $\mathfrak{n}$. I guess one can define a functor from $\mathfrak{g}$ modules to $\mathfrak{l}$ modules by $V\mapsto V_N:=V/(\mathfrak{n}.V)$. Is this a reasonable thing to do? Is there a reference where basic properties of this functor are spelled out?

The properties I am interested in is in relation to "parabolic induction" in Lie algebras. For instance, is it true that this functor is the adjoint of the functor of parabolic induction? Can one hope for an analogue of second adjointness theorem?

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The space $V / \mathfrak{n}V$ is called space of $\mathfrak{n}$-coinvariants and its $p$th left derived functor is the $p$th Lie algebra homology $H_p(\mathfrak{n},V)$.