Patrick Delorme

Constant term of smooth H_psi-spherical functions on a reductive p-adic group

Let Ψ be a smooth character of a closed subgroup, H, of a reductive p-adic group G. If P is a parabolic subgroup of G such that PH is open in G, we define the constant term of every smooth function on G which transforms by Ψ under the right action of G. The example of mixed models is given: it includes symmetric spaces and Whittaker models. In this case a notion of cuspidal function is defined and studied. It leads to finiteness theorems.

Quelques propriétés des représentations sphériques pour les espaces symétriques réductifs

Let GH be a reductive symmetric space and suppose V is an admissible (g, K)-module of finite length possessing a linear functional T ϵ Vsu which is fixed by h and H ∩ K. We prove that V can be mapped equivariantly into C∞(GH) such that T becomes the pull-back of the Dirac measure at the origin. Essential in the proof is the fact that the formal power series of certain matrix coefficients of V satisfy a system of differential equations with regular singularities.

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

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 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.

A local relative trace formula for PGL(2)

Following a scheme inspired by B. Feigon, we describe the spectral side of a local relative trace formula for

Vecteurs distributions H-invariants pour les séries principales généralisées d'espaces symétriques réductifs et prolongement méromorphe d'intégrales d'Eisenstein

G:= PGL(2,\rm E)

Classification des triples de Manin pour les algèbres de Lie réductives complexes: Avec un appendice de Guillaume Macey☆

relative to the symmetric subgroup

Meromorphic Basis of H-Invariant Distribution Vectors for the Generalized Principal Series of Reductive Symmetrical Spaces: Functional Equation

H:=PGL(2,\rm F)

Transformation de Fourier sur l'espace de Schwartz d'un espace symétrique réductif

where

Intégrales d'Eisenstein pour les espaces symétriques réductifs: Tempérance, majorations. Petite matriceB

\rm E/\rm F

Paquets d'ondes dans l'espace de Schwartz d'un espace symétrique réductif

is an unramified quadratic extension of local non archimedean fields of characteristic