Henrik Schlichtkrull

Hyperfunctions and Harmonic Analysis on Symmetric Spaces

1. Hyperfunctions and Microlocal Analysis - An Introduction.- 1.1. Hyperfunctions of one variable.- 1.2. Sheaves.- 1.3. Cohomology of sheaves.- 1.4. Hyperfunctions of several variables.- 1.5. The singular spectrum and microfunctions.- 1.6. Micro-differential operators.- 1.7. Notes.- 2. Differential Equations with Regular Singularities.- 2.1. Regular singularities for ordinary equations.- 2.2. Regular singularities for partial differential equations.- 2.3. Boundary values for a single equation.- 2.4. Example.- 2.5. Boundary values for a system of equations.- 2.6. Notes.- 3. Riemannian Symmetric Spaces and Invariant Differential Operators - Preliminaries.- 3.1. Decomposition and integral formulas for semisimple Lie groups.- 3.2. Parabolic subgroups.- 3.3. Invariant differential operators.- 3.4. Notes.- 4. A Compact Imbedding.- 4.1. Construction and analytic structure of X?.- 4.2. Invariant differential operators on X?.- 4.3. Regular singularities.- 4.4. Notes.- 5. Boundary Values and Poisson Integral Representations.- 5.1. Poisson transformations.- 5.2. Boundary value maps.- 5.3. Spherical functions and their asymptotics.- 5.4. Integral representations.- 5.5. Notes and further results.- 6. Boundary Values on the Full Boundary.- 6.1. Partial Poisson transformations.- 6.2. Partial spherical functions and Poisson kernels.- 6.3. Boundary values and asymptotics.- 6.4. The bijectivity of the partial Poisson transformations.- 6.5. Notes and further results.- 7. Semisimple Symmetric Spaces.- 7.1. The orbits of symmetric subgroups.- 7.2. Root systems.- 7.3. A fundamental family of functions.- 7.4. A differential property.- 7.5. Asymptotic expansions.- 7.6. The case of equal rank.- 7.7. Examples.- 7.8. Notes and further results.- 8. Construction ff Functions with Integrable Square.- 8.1. The invariant measure on G/H.- 8.2. An important duality.- 8.3. Discrete series.- 8.4. Examples.- 8.5. Notes and further results.

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

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.

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.

Eigenspaces of the Laplacian on hyperbolic spaces: Composition series and integral transforms

Abstract Let X be a projective real, complex, or quaternion hyperbolic space, realized as the pseudo-Riemannian symmetric space X ≅ G H with G = O(p, q), U(p, q), or Sp(p,q) (these are the classical isotropic symmetric spaces). Let Δ be the G-invariant Laplace-Beltrami operator on X. A complete description (by K-types), for each χ ∈ C , of all closed G-invariant subspaces of the eigenspace {f ∈ C ∞ (X)¦Δf = χf} is given. The eigenspace representations are compared with principal series representations, using “Poisson-transformations”. Similar results are obtained also for the exceptional isotropic symmetric space. The Langlands parameters of the spherical discrete series representations are determined.

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.

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].

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.

The Langlands parameters of Flensted-Jensen's discrete series for semisimple symmetric spaces

Abstract Let G be a semisimple Lie group. Flensted-Jensen has for certain symmetric homogeneous spaces G H given a construction of a series of irreducible unitary representations of G realized on L 2 ( G H ) . In this paper the Langlands parameters of most of these representations are found. For some sets of Langlands parameters this leads to the fact not previously known that the corresponding representations of G are actually unitary.

A Series of Unitary Irreducible Representations Induced from a Symmetric Subgroup of a Semisimple Lie Group

LetG/H be a semisimple symmetric space. Generalizing results of Flensted-Jensen we give a sufficient condition for the existence of irreducible closed invariant subspaces of the unitary representations ofG induced from unitary finite dimensional representations ofH. This provides a method of constructing unitary irreducible representations ofG, and we show by examples that for some irreducible admissible representations ofG, this method exhibits not previously known unitarity.