Let
F
F
be a locally compact field with residue characteristic
p
p
, and let
G
\mathbf {G}
be a connected reductive
F
F
-group. Let
U
\mathcal {U}
be a pro-
p
p
Iwahori subgroup of
G
=
G
(
F
)
G = \mathbf {G}(F)
. Fix a commutative ring
R
R
. If
π
\pi
is a smooth
R
[
G
]
R[G]
-representation, the space of invariants
π
U
\pi ^{\mathcal {U}}
is a right module over the Hecke algebra
H
\mathcal {H}
of
U
\mathcal {U}
in
G
G
.
Let
P
P
be a parabolic subgroup of
G
G
with a Levi decomposition
P
=
M
N
P = MN
adapted to
U
\mathcal {U}
. We complement a previous investigation of Ollivier-Vignéras on the relation between taking
U
\mathcal {U}
-invariants and various functor like
Ind
P
G
\operatorname {Ind}_P^G
and right and left adjoints. More precisely the authors’ previous work with Herzig introduced representations
I
G
(
P
,
σ
,
Q
)
I_G(P,\sigma ,Q)
where
σ
\sigma
is a smooth representation of
M
M
extending, trivially on
N
N
, to a larger parabolic subgroup
P
(
σ
)
P(\sigma )
, and
Q
Q
is a parabolic subgroup between
P
P
and
P
(
σ
)
P(\sigma )
. Here we relate
I
G
(
P
,
σ
,
Q
)
U
I_G(P,\sigma ,Q)^{\mathcal {U}}
to an analogously defined
H
\mathcal {H}
-module
I
H
(
P
,
σ
U
M
,
Q
)
I_\mathcal {H}(P,\sigma ^{\mathcal {U}_M},Q)
, where
U
M
=
U
∩
M
\mathcal {U}_M = \mathcal {U}\cap M
and
σ
U
M
\sigma ^{\mathcal {U}_M}
is seen as a module over the Hecke algebra
H
M
\mathcal {H}_M
of
U
M
\mathcal {U}_M
in
M
M
. In the reverse direction, if
V
\mathcal {V}
is a right
H
M
\mathcal {H}_M
-module, we relate
I
H
(
P
,
V
,
Q
)
⊗
c
-
I
n
d
U
G
1
I_\mathcal {H}(P,\mathcal {V},Q)\otimes \operatorname {c-Ind}_\mathcal {U}^G\mathbf {1}
to
I
G
(
P
,
V
⊗
H
M
c
-
I
n
d
U
M
M
1
,
Q
)
I_G(P,\mathcal {V}\otimes _{\mathcal {H}_M}\operatorname {c-Ind}_{\mathcal {U}_M}^M\mathbf {1},Q)
. As an application we prove that if
R
R
is an algebraically closed field of characteristic
p
p
, and
π
\pi
is an irreducible admissible representation of
G
G
, then the contragredient of
π
\pi
is
0
0
unless
π
\pi
has finite dimension.