E.P. van den Ban

The most-continuous part of the Plancherel decomposition for a reductive symmetric space

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 satis ed by the associated C functions

The principal series for a reductive symmetric space. II. Eisenstein integrals

In this paper we develop a theory of Eisenstein integrals related to the principal series for a reductive symmetric space G=H: Here G is a real reductive group of Harish-Chandras class, ? an involution of G and H an open subgroup of the group G ? of xed points for ?: The group G itself is a symmetric space for the left right action of G G : we refer to this setting as the group case. Up to a normalization, our Eisenstein integrals generalize those of Harish-Chandra [18] associated with a minimal parabolic subgroup in the group case.

A Residue Calculus for Root Systems

Let V be a finite-dimensional real vector space on which a root system Σ is given. Consider a meromorphic function ϕ on Vℂ=V+iV, the singular locus of which is a locally finite union of hyperplanes of the form λ ε Vℂ∣〈 λ, α 〉 = s, α ε Σ, s ε ℝ. Assume φ is of suitable decay in the imaginary directions, so that integrals of the form ∫η +iV ϕ λ, dλ make sense for generic η ε V. A residue calculus is developed that allows shifting η. This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.

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.

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. 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.

Analytic families of eigenfunctions on a reductive symmetric space

In harmonic analysis on a reductive symmetric space X an important role is played by families of generalized eigenfunctions for the algebra D (X) of invariant dierential operators. Such families arise for instance as matrix coeAEcients of representations that come in series, such as the (generalized) principal series. In particular, relations between such families are of great interest. We recall that a real reductive group G; equipped with the left times right multiplication action, is a reductive symmetric space. In the case of the group, examples of the mentioned relations are functional equations for Eisenstein integrals, see [23] and [25], or Arthur-Campoli relations for Eisenstein integrals, see [1], [14]. In this paper we develop a general tool to establish relations of this kind. We show that they can be derived from similar relations satised by the family of functions obtained by taking one particular coeAEcient in a certain asymptotic expansion. Since the functions in the family so obtained are eigenfunctions on symmetric spaces of lower split rank, this yields a powerful inductive method; we call it induction of relations. In the case of the group, a closely related lifting theorem by Casselman was used by Arthur in the proof of the Paley-Wiener theorem, see [1], Thm. II.4.1. However, no proof seems yet to have appeared of Casselmans theorem.

The Plancherel Theorem for a Reductive Symmetric Space

This chapter is based is on a series of lectures given at the meeting of the European School of Group Theory in August 2000, Odense, Denmark. The purpose of the lectures was to explain the structure of the Plancherel decomposition for a reductive symmetric space, as well as many of the main ideas involved in the proof found in joint work with Henrik Schlichtkrull.

Harmonic Analysis on Reductive Symmetric Spaces

We give a relatively non-technical survey of some recent advances in the Fourier theory for semisimple symmetric spaces. There are three major results: An inversion formula for the Fourier transform, a Paley—Wiener theorem, which describes the Fourier image of the space of compactly supported smooth functions, and the Plancherel theorem, which describes the decomposition into irreducibles of the regular representation.

The action of intertwining operators on spherical vectors in the minimal principal series of a reductive symmetric space

Abstract We study the action of standard intertwining operators on H-fixed generalized vectors in the minimal principal series of a reductive symmetric space G H of Harish-Chandras class. The main result is that — after an appropriate normalization — this action is unitary for the unitary principal series. This is an extension of previous work under more restrictive hypotheses on G and H. The present result implies the Maass-Selberg relations for Eisenstein integrals of the minimal principal series. These play a fundamental role in the most-continuous part of the Plancherei decomposition for G H .

Basic harmonic analysis for pseudo-Riemannian symmetric spaces

We give a survey of the present knowledge regarding basic questions in harmonic analysis on pseudo-Riemannian symmetric spaces G /H, where G is a semisimple Lie group: The definition of the Fourier transform, the Plancherel formula, the inversion formula and the Paley-Wiener theorem.