Jacquet module of irreducible principal series

Question

Let $F$ be a local field of characteristic zero and $G=\operatorname{GL}_n(F)$.

Let $B=UT$ be a Borel subgroup of $G$ and $\chi=(\chi_1,\cdots,\chi_n)$ is an unramified character of $B$.

Consider unramified principal series $\pi$ of $G$ defined by $\operatorname{Ind}_B^G \chi$. Assume that $\pi$ is irreducible.

Then for any parabolic subgroup $P_{a,n-a}=NM$ of $G$ whose Levi subgroup $M$ is $GL_{a} \times GL_{n-a}$, it is known that the any subquotient of Jacquet module $J_N(\pi)$ with respct to $N$ is also irreducible principal seires of $M$. (i.e. any subquotient of $J_N(\pi)|_{{GL_{a}}}$ and $J_N(\pi)|_{{GL_{n-a}}}$ are both principal series.)

I don't know why it holds. If it is not easy to explain, would you recommend some reference on this?

And I also wondering whether principal series representations can have finite dimensional subrepresentation.

Answer 1

I will explicitly work out the details in Section 1.2 of Zelevinsky "Induced representations of reductive $\mathfrak p$-adic groups II." In their notation, $\beta=(1,\dots,1)$ and $\gamma=(a,n-a)$. Now $I_i=\{i\}$ for each $i=1,\dots,n$ and $\mathscr I_1=\{1,\dots,a\}$ and $\mathscr I_2=\{a+1,\dots,n\}$. Now $$W^{\beta,\gamma}=\{w\in W|w^{-1}(1)<\cdots <w^{-1}(a),w^{-1}(a+1)<\cdots<w^{-1}(n)\}$$ is a subset of $W$ of size ${n\choose a}$. For each such permutation, we can define $\gamma':=\gamma\cap w(\beta)=(1,\dots,1)$ and $\beta':=\beta\cap w^{-1}(\gamma)=(1,\dots,1)$. Now Theorem 1.2 states that $$\begin{align*} J_N(\pi)^{\mathrm{s.s.}}&=\bigoplus_{w\in W^{\beta,\gamma}}i_{\gamma,\gamma'}\circ w\circ r_{\beta',\beta}(\chi_1\otimes\cdots\otimes\chi_n)\\ &=\bigoplus_{w\in W^{\beta,\gamma}}i_{\gamma,\gamma'}(\chi_{w^{-1}(1)}\otimes\cdots\otimes\chi_{w^{-1}(n)})\\ &=\bigoplus_{w\in W^{\beta,\gamma}}\mathrm{Ind}_B^{\mathrm{GL}_a}(\chi_{w^{-1}(1)}\otimes\cdots\otimes\chi_{w^{-1}(a)})\boxtimes \mathrm{Ind}_B^{\mathrm{GL}_{n-a}}(\chi_{w^{-1}(a+1)}\otimes\cdots\otimes\chi_{w^{-1}(n)}), \end{align*}$$ where as usual $\mathrm{Ind}_B^{\mathrm{GL}_a}$ denotes normalized induction. In fact, since you assume $\pi$ is irreducible, i.e., none of the $\chi_i\chi_j^{-1}$ are of the form $\nu^{\pm1}$ for any $i\ne j$, each direct summand above is irreducible, and moreover the Ext groups must vanish, so in fact the above decomposition is true before semi-simplification.

P.S. Note that none of the above used that $\chi$ is unramified.