Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments II: Minimal sequences

Abstract

Let $F$ be a non-Archimedean local field. For any irreducible smooth representation $\pi$ of $\mathrm{GL}_n(F)$ and a multisegment $\mathfrak m$, we have an operation $D_{\mathfrak m}(\pi)$ to construct a simple quotient $\tau$ of a Bernstein-Zelevinsky derivative of $\pi$. This article continues the previous one to study the following poset \[ \mathcal S(\pi, \tau) :=\left\{ \mathfrak n : D_{\mathfrak n}(\pi)\cong \tau \right\} , \] where $\mathfrak n$ runs for all the multisegments. Here the partial ordering on $\mathcal S(\pi, \tau)$ comes from the Zelevinsky ordering. We show that the poset has a unique minimal multisegment. Along the way, we introduce two new ingredients: fine chain orderings and local minimizability.