Eisenstein integrals and induction of relations

I give a survey of joint work with Henrik Schlichtkrull on the induction of certain relations among (partial) Eisenstein integrals for the minimal principal series of a reductive symmetric space. I explain the application of this principle of induction to the proofs of a Fourier inversion formula and a Paley-Wiener theorem. Finally, the relation with the Plancherel decomposition is discussed.Comment: Latex2e, 22 pp, Proc. Conf. `Analyse Harmonique Non Commutative (colloque en l'honneur de Jacques Carmona)' CIRM, Luminy, 20-24 Mai, 200

A Paley-Wiener theorem for reductive symmetric spaces

Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. The image under the Fourier transform of the space of K-finite compactly supported smooth functions on X is characterized.Comment: 31 pages, published versio

Paley-Wiener spaces for real reductive Lie groups

We show that Arthur's Paley-Wiener theorem for K-finite compactly supported smooth functions on a real reductive Lie group G of the Harish-Chandra class can be deduced from the Paley-Wiener theorem we established in the more general setting of a reductive symmetric space. In addition, we formulate an extension of Arthur's theorem to K-finite compactly supported generalized functions (distributions) on G and show that this result follows from the analogous result for reductive symmetric spaces as well.Comment: Latex2e, 28 pages, change of definition of space P^* on p. 17 + minor correction

Uniform temperedness of Whittaker integrals for a real reductive group

We study Whittaker vectors (and Jacquet integrals) in the generalized principal series for a real reductive group. A functional equation for them is obtained. This allows to establish uniform estimates for their holomorphic extensions with respect to the continuous induction parameter. Finally, we link the Whittaker vectors to Harish-Chandra's Whittaker integrals for which we then prove uniform tempered estimates. This allows us to establish rapid decay for a class of Fourier transforms on the space of Whittaker Schwartz functions.Comment: 131 page

Normalizations of Eisenstein integrals for reductive symmetric spaces

We construct minimal Eisenstein integrals for a reductive symmetric space G/H as matrix coefficients of the minimal principal series of G. The Eisenstein integrals thus obtained include those from the \sigma-minimal principal series. In addition, we obtain related Eisenstein integrals, but with different normalizations. Specialized to the case of the group, this wider class includes Harish-Chandra's minimal Eisenstein integrals

The notion of cusp forms for a class of reductive symmetric spaces of split rank one

We study a notion of cusp forms for the symmetric spaces G/H with G = SL(n,R) and H = S(GL(n-1,R) x GL(1,R)). We classify all minimal parabolic subgroups of G for which the associated cuspidal integrals are convergent and discuss the possible definitions of cusp forms. Finally, we show that the closure of the direct sum of the discrete series of representations of G/H coincides with the space of cusp forms

Transport and cooling of singly-charged noble gas ion beams

The transport and cooling of noble gas singly-charged ion beams by means of a Radio Frequency Quadrupole Cooler Buncher (RFQCB) have been studied at the LIMBE low energy beam line of the GANIL facility. Ions as light as $^{4}He^+$ have been cooled and stored before their extraction in bunches using $H_2$ as buffer gas. Bunches characteristics have been studied as a function of the parameters of the device. Sizeable transmissions of up to 10 $$ have been obtained. A detailed study of the lifetime of ions inside the buncher has been performed giving an estimate of the charge exchange cross-section. Results of a microscopic Monte-Carlo transport code show reasonable agreement with experimental data.Comment: 13 figure