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

Abstract:

Let $H$ 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 $P\cap \sigma (P)$ 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 $j_P(V)$ of $V$ along $P$ which is fixed by $M\cap H$. 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 $M\cap H$-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.