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E. P. van den Ban

Publications and source records attributed to E. P. van den Ban.

9 recordsLinked to original sources
arXiv2024

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.

math.RT↗
arXiv2011

A comparison of Paley-Wiener theorems for real reductive Lie groups

In this paper we make a detailed comparison between the Paley-Wiener theorems of J. Arthur and P. Delorme. We prove that these theorems are equivalent from an a priori point of view. We also give an alternative formulation of the theorems in terms of the Hecke algebra of bi-K-finite distributions supported on K. Our techniques involve derivatives of holomorphic families of continuous representations and Harish-Chandra modules.

math.RT↗
arXiv2005

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.

math.RT↗
arXiv2004

The Plancherel decomposition for a reductive symmetric space I. Spherical functions

We prove the Plancherel formula for spherical Schwartz functions on a reductive symmetric space. Our starting point is an inversion formula for spherical smooth compactly supported functions. The latter formula was earlier obtained from the most continuous part of the Plancherel formula by means of a residue calculus. In the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized Eisenstein integrals and of the Maass-Selberg relations satisfied by the associated C-functions.

math.RT↗
arXiv2004

The Plancherel decomposition for a reductive symmetric space II. Representation theory

We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I (math.RT/0107063). The formula for Schwartz functions involves Eisenstein integrals obtained by a residual calculus. In the present paper we identify these integrals as matrix coefficients of the generalized principal series.

math.RT↗
arXiv2003

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.

math.RT↗
arXiv2001

Analytic families of eigenfunctions on a reductive symmetric space

The asymptotic behavior of holomorphic families of generalized eigenfunctions on a reductive symmetric space is studied. The family parameter is a complex character on the split component of a parabolic subgroup. The main result asserts that the family vanishes if a particular asymptotic coefficient does. This allows an induction of relations between families that will be applied in forthcoming work on the Plancherel and the Paley-Wiener theorem.

math.RT↗