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

Abstract

Abstract. Let H be the fixed point group of a rational involution σ of a reductive p-adic group on a field of characteristic different from 2. Let P be a σ-parabolic subgroup of G i.e. such that σ(P ) is opposite to P . We denote the intersection P ∩ σ(P ) by M . Kato and Takano on one hand, Lagier on the other hand associated canonically to an H-form, i.e. an H-fixed linear form, ξ, 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 ∩ H. We call this operation constant term of H-forms. This constant term is linked to the asymptotic behaviour of the generalized coefficients with respect to ξ.P. Blanc and the second author defined a family of H-forms on certain parabolically induced representations, associated to an M ∩ H-form, η, 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 η is discrete, i.e. when the generalized coefficients of η 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, of H-forms. A formula for the constant term of Eisenstein integrals is given.

The support of closed orbit relative matrix coefficients

“…It is natural to ask about non-vanishing of the invariant forms λ N,χ where P = M N is a θ-split parabolic subgroup, ρ is M θ -distinguished, and λ ∈ Hom H (ι G P ρ, χ) arises from the open orbit in P \G/H (such linear functionals were constructed and studied in [BD08]). This question has been addressed by Carmona and Delorme in [CD14] and we refer the reader to their account.…”

Isomorphy Classes of $k$-Involutions of $\text{SO}(n, k,β)$, $n > 2$

“…Symmetric spaces play an important role in many areas of mathematics and physics, but probably best known are the representations associated with these symmetric spaces which have been studied by many prominent mathematicians starting with a study of compact groups and their representations by Cartan [Car29], to a study of Riemannian symmetric spaces and real (and padic) groups by Harish Chandra [HC84] to a more recent study of the non Riemannian symmetric spaces (see for example [Far79,FJ80,ŌS80,vdBS97b,vdBS97a]) leading to a Plancherel formula in 1996 by Delorme [Del98]. Once this Plancherel formula was obtained the attention shifted to p-adic symmetric spaces (see for example [DH14,CD14,Del13,HH02,HH05]). In the late 1980's generalizations of these reductive symmetric spaces to other base fields started to play a role in other areas, like in the study of arithmetic subgroups (see [TW89]), the study of character sheaves (see for example [Lus90,Gro92]), geometry (see [DCP83,DCP85] and [Abe88]), singularity theory (see [LV83] and [HS90]), and the study of Harish Chandra modules (see [BB81] and [Vog83,Vog82]).…”