Erik P. van den Ban

Fourier inversion on a reductive symmetric space

Let X be a semisimple symmetric space. In previous papers, [8] and [9], we have dened an explicit Fourier transform for X and shown that this transform is injective on the space C 1 c (X) ofcompactly supported smooth functions on X. In the present paper, which is a continuation of these papers, we establish an inversion formula for this transform.

Expansions for Eisenstein integrals on semisimple symmetric spaces

1. Introduction Let G/H be a semisimple symmetric space. Related to the (minimal) principal series for G/H there is a series of Eisenstein integrals on G/H. These are K-finite joint eigenfunctions for the G-invariant differential operators on G/H. Here K is a maximal compact subgroup of G. The Eisenstein integrals are generalizations of the elementary spherical functions for a Riemannian symmetric space (and more generally of the generalized spherical functions in [9, w and of Harish-Chandras Eisenstein integrals associated to a minimal parabolic subgroup of a semisimple Lie group. In this paper we develop a theory of asymptotic (in fact, converging) expansions towards infinity for the Eisenstein integrals. The theory generalizes Harish-Chandras theory (see [8, Thm. IV.5.5], and [13, Thm. 9.1.5.1]) in the two cases mentioned above (see also [9, Thm. III.2.7]). The main results are Theorems 9.1 and 11.1. The first of these states the convergence on an open Weyl chamber of the series expansion whose coefficients are derived recursively from the differential equations satisfied by the Eisenstein integrals. The sum Oh of the series is an eigen-function which behaves regularly at infinity but in general is singular at the walls of the chamber. The basic estimates which ensure the convergence of the series also provide an estimate for ~x, which is a generalization of Gangollis estimates ([7]) in the Riemannian case. As in Gangollis case, our estimates are derived by a modification of the Oh with the square root of a certain Jacobian function. The second main result expresses the Eisenstein integral as a linear combination of the Oh; the coefficients are the c-functions (defined in previous work by one of us) related to the Eisenstein integrals. The results of this paper are used for the Plancherel and Paley-Wiener type results obtained in [5] for the Fourier transform corresponding to the minimal prin-60 Erik P. van den Ban and Henrik Schlichtkrull cipal series, just as Gangollis estimates in the Riemannian case play a crucial role in Helgasons and Rosenbergs work for the spherical transform (see [8, w In the case of a semisimple Lie group, considered as a symmetric space, estimates sufficient for the application to the Paley-Wiener theorem are given in [1]. The present, stronger, estimates were in this case obtained in [6].

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.

Symplectic geometry of semisimple orbits

Abstract Let G be a complex semisimple group, T ⊂ G a maximal torus and B a Borel subgroup of G containing T . Let Ω be the Kostant-Kirillov holomorphic symplectic structure on the adjoint orbit O = Ad( G ) c ≃ G / Z ( c ), where c ∈ Lie( T ), and Z ( c ) is the centralizer of c in G . We prove that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad( c ) are real (respectively, purely imaginary). Furthermore, each of these real symplectic manifolds is symplectomorphic to the cotangent bundle of the partial flag manifold G/Z ( c c) B , equipped with the Liouville symplectic form. The latter result is generalized to hyperbolic adjoint orbits in a real semisimple Lie algebra.

Paley-Wiener spaces for real reductive Lie groups

We show that Arthurs 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 Arthurs 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.

Asymptotic Expansions on Symmetric Spaces

Let G/H be a semisimple symmetric space, where G is a connected semisimple real Lie group with an involution σ, and H is an open subgroup of the fix point group Gσ. Assume that G has finite center; then it is known that G has a σ-stable maximal compact subgroup K.