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

Constant term of Eisenstein integrals on a reductive 𝑝-adic symmetric space