4

Let $G$ be (the rational points of) a connected, reductive group over a local field $F$. Let $P$ be a parabolic subgroup of $G$ with unipotent radical $N$ and Levi subgroup $M$. The inclusion $M \subseteq P$ induces an isomorphism of $M$ with $P/N$. Let $\iota$ be the inclusion of $P$ into $G$, and $\rho$ the composition of the projection $P \rightarrow P/N$ and the isomorphism $P/N \xrightarrow{\simeq} M$.

Let $(V,\pi)$ be a smooth representation of $G$. The Jacquet module $J_{G,M}(V)$ is the quotient $V/V(N)$, where $V(N)$ is the subspace of $V$ spanned by $\pi(n)v - v$. It inherits the structure of a smooth representation of $M$, since $V(N)$ is $M$-stable.

The article Representations of $\mathfrak p$-adic groups, Pierre Cartier, Corvallis Proceedings I, gives another definition of $J_{G,M}(V)$.

enter image description here

Here $\mathscr S_G$ is the category of smooth representations of $G$. As indicated, $\iota^{\ast}$ is the restriction functor. The definition of $\rho_{\ast}$ is more complicated: a smooth representation of a group $H$ is the same thing as a nondegenerate $\mathscr H(H)$-module, where $\mathscr H(H)$ is the Hecke algebra of $H$. The map $\rho: P \rightarrow M$ gives $\mathscr H(M)$ the structure of a right $\mathscr H(P)$-module, and one then defines

$$\rho_{\ast}(V) = \mathscr H(M) \otimes_{\mathscr H(P)} V$$

as an $\mathscr H(M)$-module, hence as a smooth representation of $M$.

This definition of the functor $\rho_{\ast}$ is so abstract and I don't have any intuition on how to work with it. How should one think about this abstract definition of $J_{G,M}$, in order to relate it to the earlier definition?

D_S
  • 35,843

1 Answers1

1

The universal property of the Jacuqet functor $\pi\mapsto \pi_N$ is that for any $M$-representation $\varphi$ we have $\hom_G(\pi,\mathrm{Ind}_P^G\varphi)\cong \hom_M(\pi_N,\varphi)$.

I will check $J_{G,M}=\rho_*\iota^*$ satisfies this property. Indeed, $$\begin{align*} \hom_M(\rho_*\iota^*\pi,\varphi)&=\hom_{\mathcal H(M)}(\iota^*\pi\otimes_{\mathcal H(P)}\mathcal H(M),\varphi)\\ &=\hom_{\mathcal H(P)}(\iota^*\pi,\varphi)\\ &=\hom_{\mathcal H(G)}(\pi,\mathrm{Ind}_P^G\varphi). \end{align*}$$ Here, the second equality is the universal property of the tensor product, and the last equality is Frobenius reciprocity.

Kenta S
  • 18,181