Let
G
\mathbf {G}
be a connected reductive group defined over a locally compact non-archimedean field
F
F
, let
P
\mathbf {P}
be a parabolic subgroup with Levi
M
\mathbf {M}
and compatible with a pro-
p
p
Iwahori subgroup of
G
≔
G
(
F
)
G ≔\mathbf {G}(F)
. Let
R
R
be a commutative unital ring.
We introduce the parabolic pro-
p
p
Iwahori–Hecke
R
R
-algebra
H
R
(
P
)
\mathcal {H}_R(P)
of
P
≔
P
(
F
)
P ≔\mathbf {P}(F)
and construct two
R
R
-algebra morphisms
Θ
M
P
:
H
R
(
P
)
→
H
R
(
M
)
\Theta ^P_M\colon \mathcal {H}_R(P)\to \mathcal {H}_R(M)
and
Ξ
G
P
:
H
R
(
P
)
→
H
R
(
G
)
\Xi ^P_G\colon \mathcal {H}_R(P) \to \mathcal {H}_R(G)
into the pro-
p
p
Iwahori–Hecke
R
R
-algebra of
M
≔
M
(
F
)
M ≔\mathbf {M}(F)
and
G
G
, respectively. We prove that the resulting functor
Mod
-
H
R
(
M
)
→
Mod
-
H
R
(
G
)
\operatorname {Mod}\text {-}\mathcal {H}_R(M) \to \operatorname {Mod}\text {-}\mathcal {H}_R(G)
from the category of right
H
R
(
M
)
\mathcal {H}_R(M)
-modules to the category of right
H
R
(
G
)
\mathcal {H}_R(G)
-modules (obtained by pulling back via
Θ
M
P
\Theta ^P_M
and extension of scalars along
Ξ
G
P
\Xi ^P_G
) coincides with the parabolic induction due to Ollivier–Vignéras.
The maps
Θ
M
P
\Theta ^P_M
and
Ξ
G
P
\Xi ^P_G
factor through a common subalgebra
H
R
(
M
,
G
)
\mathcal {H}_R(M,G)
of
H
R
(
G
)
\mathcal {H}_R(G)
which is very similar to
H
R
(
M
)
\mathcal {H}_R(M)
. Studying these algebras
H
R
(
M
,
G
)
\mathcal {H}_R(M,G)
for varying
(
M
,
G
)
(M,G)
we prove a transitivity property for tensor products. As an application we give a new proof of the transitivity of parabolic induction.