On mod $p$ local-global compatibility for ${\mathrm{G}\mathrm{L}}_n({\mathbf{Q}}_p)$ in the ordinary case

Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation $\bar{r}:\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbf{Q}}\mathrm{/}F)\to {\mathrm{G}\mathrm{L}}_n({\bar{\mathbf{F}}}_p)$ is upper-triangular and satisfies certain genericity conditions at a place $w$ above $p$, and that every subquotient of $\bar{r}{\mathrm{\mid }}_{\mathrm{G}\mathrm{a}\mathrm{l}({\bar{\mathbf{Q}}}_p\mathrm{/}{F}_w)}$ of dimension $>2$ is Fontaine–Laffaille generic. In this paper, we show that the isomorphism class of $\bar{r}{\mathrm{\mid }}_{\mathrm{G}\mathrm{a}\mathrm{l}({\bar{\mathbf{Q}}}_p\mathrm{/}{F}_w)}$ is determined by ${\mathrm{G}\mathrm{L}}_n(F_w)$-action on a space of mod $p$ algebraic automorphic forms cut out by the maximal ideal of a Hecke algebra associated to $\bar{r}$, assuming a weight elimination result which is a theorem of Bao V. Le Hung in his forthcoming paper~\cite{LeH}. In particular, we show that the wildly ramified part of $\bar{r}{\mathrm{\mid }}_{\mathrm{G}\mathrm{a}\mathrm{l}({\bar{\mathbf{Q}}}_p\mathrm{/}{F}_w)}$ is determined by the action of Jacobi sum operators (seen as elements of ${\mathbf{F}}_p[{\mathrm{G}\mathrm{L}}_n({\mathbf{F}}_p)]$) on this space.