Mod p local-global compatibility for GSp₄(Qₚ) in the ordinary case

Let F be a totally real field of even degree in which p splits completely. Let r̄:G_F_4(F̄_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 GSp₄(Fw) -action on the corresponding Hecke-isotypic part of the space of mod p automorphic forms on a compact mod center form of GSp₄ with infinite level at w determines r̄|_{GF_w} .