  Let G be the group of rational points of a reductive connected group over a
finite field (resp. nonarchimedean local field of characteristic p) and R a
commutative ring. The unipotent (resp. pro-p Iwahori) invariant functor takes a
smooth representation of G to a module over the unipotent (resp. pro-p Iwahori)
Hecke R-algebra H of G. We prove that these functors for G and for a Levi
subgroup of G commute with the parabolic induction functors, as well as 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 in the
local case that an irreducible admissible R-representation V of G is
supercuspidal (or equivalently supersingular) if and only if the H-module V^I
of its invariants by the pro-p Iwahori I admits a supersingular subquotient, if
and only if V^I is supersingular.
