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 $π$ of $\mathrm{GL}n(F)$ and a multisegment $\mathfrak m$, we have an operation $D{\mathfrak m}(π)$ to construct a simple quotient $τ$ of a Bernstein-Zelevinsky derivative of $π$. This article continues the previous one to study the following poset [ \mathcal S(π, τ) :=\left{ \mathfrak n : D_{\mathfrak n}(π)\cong τ\right} , ] where $\mathfrak n$ runs for all the multisegments. Here the partial ordering on $\mathcal S(π, τ)$ 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.