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If $(\pi,V)$ is a representation of $G=GL_n(F)$ where $F$ is a nonarchimedean local field, and $0 \subset V_2 \subset V_1 \subset V$ is a filtration of $V$ into $G$-invariant subspaces, with $V/V_1$ and $V_1/V_2$ supercuspidal, must $\pi$ be supercuspidal?

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For any parabolic $P = MN$ (where $N$ denotes the unipotent radical of $P$) of $G$ we have the Jacquet functor $r_{M,G}$ mapping a $G$-module $V$ to the $M$-module $V/\langle n v - v \:|\: n \in N, v \in V \rangle$, the space of $N$-coinvariants and $V$ is supercuspidal if $r_{M,G}(V) = 0$ for all $P = MN$.

It is fairly easy to see that the Jacquet functors are exact, so if $V/V_1$ and $V_1/V_2$ vanish under $r_{M,G}$, so does $V/V_2$. To conclude that $V$ is supercuspidal, you will however have to assume that not only $V/V_1$ and $V_1/V_2$ are supercuspidal, but that $V_2$ is also supercuspidal.

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