Representations of a p-adic group in characteristic p

Let F be a locally compact non-archimedean field of residue characteristic p, G a connected reductive group over F, and R a field of characteristic p. When R is algebraically closed, the irreducible admissible R representations of G = G(F) are classified in [J. Amer. Math. Soc. 30 (2017), no. 2, 495-559] in term of supersingular R-representations of the Levi subgroups of G and parabolic induction; there is a similar classification for the simple modules of the pro-p Iwahori Hecke R-algebra H(G)(R) in [N. Abe, DOI:10.1515/crelle-2016-0043]. In this paper, we show that both classifications hold true when R is not algebraically closed.