Injectivity conjecture for minimal multisegment derivatives

Let $G_n$ be the relevant general linear group, let $\rho$ be the fixed cuspidal representation, and let $\mathrm{Irr}_{\rho}$ and $\mathrm{Mult}_{\rho}$ denote the corresponding irreducible representations and multisegments. For $\mathfrak n=\{\Delta_1,\ldots,\Delta_r\}\in\mathrm{Mult}_{\rho}$, write $D_{\mathfrak n}$ for the associated iterated derivative, $\widetilde{\lambda}(\mathfrak n)=\mathrm{St}(\Delta_1)\times\cdots\times\mathrm{St}(\Delta_r)$ for its co-standard representation, and $l_{\mathrm{abs}}(\Delta_i)$ for the absolute length of $\Delta_i$. A multisegment is minimal to $\pi$ when it has the minimality property defined in the paper.

Generalized injectivity conjecture. Let $\pi\in\mathrm{Irr}_{\rho}$ and let $\mathfrak n\in\mathrm{Mult}_{\rho}$ be minimal to $\pi$. Set

$$l=l_{\mathrm{abs}}(\Delta_1)+\cdots+l_{\mathrm{abs}}(\Delta_r).$$

Then the unique nonzero map

$$D_{\mathfrak n}(\pi)\boxtimes\widetilde{\lambda}(\mathfrak n)\longrightarrow\pi_{N_l}$$

is injective.

This conjecture generalizes the injectivity result for a pair of linked segments to arbitrary minimal multisegments; its resolution is intended to clarify the embedding model for Bernstein–Zelevinsky derivatives.