# 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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Yes, this is indeed a reasonable thing to do.

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 $H_p(\mathfrak{n},V)$.

The most comprehensive reference would be Knapp, Vogan: Cohomological Induction and Unitary Representations.

Edited:

By superficially skimming over the book by Knapp & Vogan, I would be inclined to say that the second adjointness theorem is true. I am certainly far from being an expert in these matters. If this book is too big for you I can recommend Dirac Operators in Representation Theory by Pandzic and Huang and its two chapters on cohomological induction.

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Yes, these ideas have definitely been echoed in Lie algebra representations. Beyond this, the work of Joseph and others on enveloping algebras and their primitive ideals goes further in the algebraic direction. I think Victor Ginzburg has even worked out similar ideas for Cherednik algebras. The moral seems to be that basic insights anywhere in representation theory tend to cross boundaries over time (thoough not always with equal success). –  Jim Humphreys Oct 17 '13 at 17:37