Let
H
H
be the fixed point group of a rational involution
σ
\sigma
of a reductive
p
p
-adic group on a field of characteristic different from 2. Let
P
P
be a
σ
\sigma
-parabolic subgroup of
G
G
, i.e. such that
σ
(
P
)
\sigma (P)
is opposite
P
P
. We denote the intersection
P
∩
σ
(
P
)
P\cap \sigma (P)
by
M
M
.
Kato and Takano on one hand and Lagier on the other associated canonically to an
H
H
-form, i.e. an
H
H
-fixed linear form,
ξ
\xi
, on a smooth admissible
G
G
-module,
V
V
, a linear form on the Jacquet module
j
P
(
V
)
j_P(V)
of
V
V
along
P
P
which is fixed by
M
∩
H
M\cap H
. We call this operation the constant term of
H
H
-forms. This constant term is linked to the asymptotic behaviour of the generalized coefficients with respect to
ξ
\xi
.
P. Blanc and the second author defined a family of
H
H
-forms on certain parabolically induced representations, associated to an
M
∩
H
M\cap H
-form,
η
\eta
, on the space of the inducing representation.
The purpose of this article is to describe the constant term of these
H
H
-forms.
Also it is shown that when
η
\eta
is discrete, i.e. when the generalized coefficients of
η
\eta
are square integrable modulo the center, the corresponding family of
H
H
-forms on the induced representations is a family of tempered, in a suitable sense,
H
H
-forms. A formula for the constant term of Eisenstein integrals is given.