Parabolic induction in characteristic p

Let F (resp. F) be a nonarchimedean locally compact field with residue characteristic p (resp. a finite field with characteristic p). For k = F or k = F, let G be a connected reductive group over k and R be a commutative ring. We denote by Rep(G(k)) the category of smooth R-representations of G(k). To a parabolic k-subgroup P = MN of G corresponds the parabolic induction functor Ind(P(k))(G(k)) : Rep(M(k)) -> Rep(G(k)). This functor has a left and a right adjoint. Let U (resp. U) be a pro-p Iwahori (resp. a p-Sylow) subgroup of G(k) compatible with P(k) when k = F (resp. F). Let H-G(k) denote the pro-p Iwahori (resp. unipotent) Hecke algebra of G(k) over R and Mod(H-G(k)) the category of right modules over HG(k). There is a functor Ind(HM(k))(HG(k)) : Mod(H-M(k)) -> Mod(H-G(k)) called parabolic induction for Hecke modules; it has a left and a right adjoint. We prove that the pro-p Iwahori (resp. unipotent) invariant functors commute with the parabolic induction functors, namely that Ind(P(k))(G(k)) and Ind(HM(k))(HG(k)) form a commutative diagram with the U and U boolean AND M(F) (resp. U and Un M(F)) invariant functors. We prove that the pro-p Iwahori (resp. unipotent) invariant functors also commute with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction functors in general; they do if p is invertible in R. When R is an algebraically closed field of characteristic p, we show that an irreducible admissible R-representation of G(F) is supercuspidal (or equivalently supersingular) if and only if the H-G(F)-module m of its U-invariants admits a supersingular subquotient, if and only if m is supersingular.