Jacquet module - Knowlpedia

Definition

Let $F$ be a nonarchimedean local field, let $G$ be a connected reductive $F$ -group, and write $P=MN$ for a parabolic subgroup with Levi subgroup $M$ and unipotent radical $N$ . For a smooth representation $(\pi,V)$ of $G(F)$ , its unnormalized Jacquet module along $P$ is the $N(F)$ -coinvariant space

V_N=V/\langle\pi(n)v-v:n\in N(F),\ v\in V\rangle.

It is naturally a smooth representation of $M(F)$ . The normalized Jacquet module is

r_P^G(V)=\delta_P^{-1/2}\otimes V_N,

where $\delta_P$ is the parabolic modulus character.

Adjointness

With compatible normalizations, normalized parabolic induction $i_P^G$ is left adjoint to $r_P^G$ . Bernstein's second adjointness theorem gives $r_{\overline P}^G$ , for the opposite parabolic $\overline P$ , as a left adjoint to $i_P^G$ .

An irreducible admissible representation is supercuspidal exactly when all of its Jacquet modules for proper parabolic subgroups vanish.

Exactness

Over complex coefficients the Jacquet functor is exact on smooth representations of a reductive $p$ -adic group. This is substantially stronger than arbitrary coinvariants and is one reason it is central to the structure theory of parabolic induction.

References

I. N. Bernstein and A. V. Zelevinsky, “Induced representations of reductive $p$ -adic groups. I,” Annales scientifiques de l'École Normale Supérieure 10 (1977), 441–472. Numdam.
William Casselman, “Introduction to the theory of admissible representations of $p$ -adic reductive groups,” unpublished notes, §4. UBC.