Constant term of $H$-forms

TL;DR

This paper studies the constant term of $H$-fixed linear forms on representations of reductive $p$-adic groups, extending previous work to more general settings and analyzing their asymptotic behavior and temperedness.

Contribution

It describes the constant term of $H$-fixed linear forms on induced representations and shows their temperedness when associated to square integrable forms.

Findings

The constant term relates to the asymptotic behavior of generalized coefficients.

When $ ext{eta}$ is square integrable, the $H$-fixed linear forms are tempered.

The work extends previous results by removing restrictions on the residue field characteristic.

Abstract

Let $H$ be the fixed point group of a rational involution $\si$ of a reductive $p$-adic group of charactersistic different from 2(this new version allows to remove the hypothesis on the characteristic of the residue field, see Proposition 2.3 and section 10). Let $P$ be a $\si$-parabolic subgroup of $G $ i.e. such that $\si(P)$ is opposite to $P$. We denote by $M$ the intersection with $\si(P)$. Kato and Takano on one hand, Lagier on the other hand 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 constant term of $H$-fixed linear 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 $H$-fixed linear forms on certain parabolically induced representations, associated to an $M\cap H$-fixed linear form, $\eta$, on the space of the inducing representation. The purpose of this article is to describe the constant term of these $H$-fixed linear forms. Also it is shown that when $\eta$ is square integrable, i.e. when the generalized coefficients of $\eta$ are square integrable, the corresponding family of $H$-fixed linear forms on the induced representations is a family of tempered, in a suitable sense, of $H$-fixed linear forms.