Mod $p$ local-global compatibility for ${\mathrm{GSp}}_4({\mathbb{Q}}_p)$ in the ordinary case

Abstract

Let $F$ be a totally real field of even degree in which $p$ splits completely. Let $\bar{r}:G_F\to {\mathrm{GSp}}_4({\overline{\mathbb{F}}}_p)$ be a modular Galois representation unramified at all finite places away from $p$ and upper-triangular, maximally nonsplit, and of parallel weight at places dividing $p$. Fix a place $w$ dividing $p$. Assuming certain genericity conditions and Taylor–Wiles assumptions, we prove that the ${\mathrm{GSp}}_4(F_w)$-action on the corresponding Hecke-isotypic part of the space of mod $p$ automorphic forms on a compact mod center form of ${\mathrm{GSp}}_4$ with infinite level at $w$ determines $\bar{r}{|}_{G_{F_w}}$.