K-finite joint eigenfunctions of U(g)K on a non-riemannian semisimple symmetric space G/H

1981 ◽  
pp. 91-101 ◽  
Author(s):  
Mogens Flensted-Jensen
Keyword(s):  
Symmetric Space ◽  
Semisimple Symmetric Space ◽  
Joint Eigenfunctions

Harmonic Spinors on Hyperbolic Space

1993 ◽  
Vol 36 (3) ◽  
pp. 257-262 ◽  
Author(s):  
Pierre-Yves Gaillard
Keyword(s):  
Differential Equations ◽  
Symmetric Space ◽  
Representation Theory ◽  
Hyperbolic Space ◽  
Harmonic Functions ◽  
Short Note ◽  
Harmonic Forms ◽  
Semisimple Symmetric Space ◽  
Invariant Differential ◽  
Harmonic Spinors

AbstractThe purpose for this short note is to describe the space of harmonic spinors on hyperbolicn-spaceHn. This is a natural continuation of the study of harmonic functions onHnin [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms onHnwere considered. The connection between invariant differential equations on a Riemannian semisimple symmetric spaceG/Kand homological aspects of the representation theory ofG, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

Fourier transforms on a semisimple symmetric space

1997 ◽  
Vol 130 (3) ◽  
pp. 517-574 ◽  
Author(s):  
Erik van den Ban ◽  
Henrik Schlichtkrull
Keyword(s):  
Symmetric Space ◽  
Fourier Transforms ◽  
Semisimple Symmetric Space

scholarly journals Eigenspaces of the Laplace-Beltrami operator on a hyperboloid

1980 ◽  
Vol 79 ◽  
pp. 151-185 ◽  
Author(s):  
Jiro Sekiguchi
Keyword(s):  
Differential Equations ◽  
Symmetric Space ◽  
Differential Operators ◽  
Beltrami Operator ◽  
Poisson Integral ◽  
Riemannian Symmetric Space ◽  
Systems Of Differential Equations ◽  
Rank One ◽  
Joint Eigenfunctions ◽  
Invariant Differential

Ever since S. Helgason [4] showed that any eigenfunction of the Laplace-Beltrami operator on the unit disk is represented by the Poisson integral of a hyperfunction on the unit circle, much interest has been arisen to the study of the Poisson integral representation of joint eigenfunctions of all invariant differential operators on a symmetric space X. In particular, his original idea of expanding eigenfunctions into K-finite functions has proved to be generalizable up to the case where X is a Riemannian symmetric space of rank one (cf. [4], [5], [11]). Presently, extension to arbitrary rank has been completed by quite a different formalism which views the present problem as a boundary-value problem for the differential equations. It should be recalled that along this line of approach a general theory of the systems of differential equations with regular singularities was successfully established by Kashiwara-Oshima (cf. [6], [7]).

scholarly journals Some results on Semisimple Symmetric Spaces and Invariant Differential Operators

2017 ◽  
Vol 116 (2) ◽  
Author(s):  
Trần Đạo Dõng
Keyword(s):  
Symmetric Space ◽  
Open Subset ◽  
Symmetric Spaces ◽  
Differential Operators ◽  
Analytic Manifold ◽  
Invariant Differential Operators ◽  
Semisimple Symmetric Space ◽  
Real Analytic Manifold ◽  
Invariant Differential ◽  
Real Analytic

<pre>Let X = G/H be a semisimple symmetric space of non-compact style. Our purpose is to construct a compact real analytic manifold in which the semisimple symmetric space X = G/H is realized as an open subset and that $G$ acts analytically on it.</pre><pre> By the <span>Cartan</span> decomposition <span>G = KAH,</span> we must <span>compacify</span> the <span>vectorial</span> part <span>A.$</span></pre><pre> In [6], by using the action of the Weyl group, we constructed a compact real analytic manifold in which the semisimple symmetric space G/H is realized as an open subset and that G acts analytically on it.</pre><pre>Our construction is a motivation of the <span>Oshima's</span> construction and it is similar to those in N. <span>Shimeno</span>, J. <span>Sekiguchi</span> for <span>semismple</span> symmetric spaces.</pre><pre>In this note, first we will <span>inllustrate</span> the construction via the case of <span>SL (n, </span>R)/SO_e (1, n-1) and then show that the system of invariant differential operators on X = G/H extends analytically on the corresponding compactification. </pre>

Homogeneous space with non-virtually abelian discontinuous groups but without any proper SL(2, ℝ)-action

2016 ◽  
Vol 27 (03) ◽  
pp. 1650018
Author(s):  
Takayuki Okuda
Keyword(s):  
Symmetric Space ◽  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Powerful Method ◽  
Discrete Subgroups ◽  
Discontinuous Group ◽  
Semisimple Symmetric Space ◽  
Continuous Analogue ◽  
Discontinuous Groups

In the study of discontinuous groups for non-Riemannian homogeneous spaces, the idea of “continuous analogue” gives a powerful method (T. Kobayashi [Math. Ann. 1989]). For example, a semisimple symmetric space [Formula: see text] admits a discontinuous group which is not virtually abelian if and only if [Formula: see text] admits a proper [Formula: see text]-action (T. Okuda [J. Differ. Geom. 2013]). However, the action of discrete subgroups is not always approximated by that of connected groups. In this paper, we show that the theorem cannot be extended to general homogeneous spaces [Formula: see text] of reductive type. We give a counterexample in the case [Formula: see text].

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