An application of shift operators to ordered symmetric spaces

Nils Byrial Andersen[1]; Jérémie M. Unterberger[2]

  • [1] University of New South Wales, School of Mathematics, Sydney NSW 2052 (Australie)
  • [2] Université Henri Poincaré, Institut Élie Cartan, BP 239, 54506 Vandœuvre-lès-Nancy Cedex (France)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 1, page 275-288
  • ISSN: 0373-0956

Abstract

top
We study the action of elementary shift operators on spherical functions on ordered symmetric spaces ℳ m , n of Cayley type, where m ∈ ℕ denotes the multiplicity of the short roots and n ∈ ℕ the rank of the symmetric space. For m even we apply this to prove a Paley-Wiener theorem for the spherical Laplace transform defined on ℳ m , n by a reduction to the rank 1 case. Finally we generalize our notions and results to B C n type root systems.

How to cite

top

Andersen, Nils Byrial, and Unterberger, Jérémie M.. "An application of shift operators to ordered symmetric spaces." Annales de l’institut Fourier 52.1 (2002): 275-288. <http://eudml.org/doc/115977>.

@article{Andersen2002,
abstract = {We study the action of elementary shift operators on spherical functions on ordered symmetric spaces $\{\mathcal \{M\}\}_\{m,n\}$ of Cayley type, where $m\in \{\mathbb \{N\}\}$ denotes the multiplicity of the short roots and $n\in \{\mathbb \{N\}\}$ the rank of the symmetric space. For $m$ even we apply this to prove a Paley-Wiener theorem for the spherical Laplace transform defined on $\{\mathcal \{M\}\}_\{m,n\}$ by a reduction to the rank 1 case. Finally we generalize our notions and results to $BC_n$ type root systems.},
affiliation = {University of New South Wales, School of Mathematics, Sydney NSW 2052 (Australie); Université Henri Poincaré, Institut Élie Cartan, BP 239, 54506 Vandœuvre-lès-Nancy Cedex (France)},
author = {Andersen, Nils Byrial, Unterberger, Jérémie M.},
journal = {Annales de l’institut Fourier},
keywords = {ordered symmetric spaces; spherical Laplace transform; shift operators; Cayley spaces; spaces of Cayley type; Paley-Wiener theorem; wave packets},
language = {eng},
number = {1},
pages = {275-288},
publisher = {Association des Annales de l'Institut Fourier},
title = {An application of shift operators to ordered symmetric spaces},
url = {http://eudml.org/doc/115977},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Andersen, Nils Byrial
AU - Unterberger, Jérémie M.
TI - An application of shift operators to ordered symmetric spaces
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 1
SP - 275
EP - 288
AB - We study the action of elementary shift operators on spherical functions on ordered symmetric spaces ${\mathcal {M}}_{m,n}$ of Cayley type, where $m\in {\mathbb {N}}$ denotes the multiplicity of the short roots and $n\in {\mathbb {N}}$ the rank of the symmetric space. For $m$ even we apply this to prove a Paley-Wiener theorem for the spherical Laplace transform defined on ${\mathcal {M}}_{m,n}$ by a reduction to the rank 1 case. Finally we generalize our notions and results to $BC_n$ type root systems.
LA - eng
KW - ordered symmetric spaces; spherical Laplace transform; shift operators; Cayley spaces; spaces of Cayley type; Paley-Wiener theorem; wave packets
UR - http://eudml.org/doc/115977
ER -

References

top
  1. N.B. Andersen, G. Ólafsson, Paley-Wiener Theorems for the Spherical Laplace transform on Causal Symmetric Spaces of rank 1, Proc. Amer. Math. Soc. 129 (2001), 173-179 Zbl0958.43004MR1695108
  2. N.B. Andersen, J. M. Unterberger, Harmonic Analysis on S U ( n , n ) / S L ( n , ℂ ) × ℝ + * , J. Lie Theory 10 (2000), 311-322 Zbl0961.22011MR1774862
  3. J. Faraut, Fonctions sphériques sur un espace symétrique ordonné de type Cayley, Contemp. Math. 191 (1995), 41-55 Zbl0847.53039MR1365533
  4. J. Faraut, J. Hilgert, G. Ólafsson, Spherical functions on ordered symmetric spaces, Ann. Inst. Fourier 44 (1994), 927-966 Zbl0810.43003MR1303888
  5. G. Heckman, H. Schlichtkrull, Harmonic analysis and special functions on symmetric spaces, (1994), Academic Press Zbl0836.43001MR1313912
  6. G. Heckman, Dunkl Operators, Séminaire Bourbaki Vol. 1996/97 ; No 245 ; Exp. No 828, 4 (1997), 223-246 Zbl0916.33012
  7. S. Helgason, Groups and Geometric Analysis, (1984), Academic Press, Orlando Zbl0543.58001MR754767
  8. M. Mizony, Une transformation de Laplace-Jacobi, SIAM J. Math. Anal. 14 (1983), 987-1003 Zbl0519.44003MR711179
  9. G. Ólafsson, Symmetric spaces of Hermitian type, Differential Geom. Appl. 1 (1991), 195-233 Zbl0785.22021MR1244444
  10. G. Ólafsson, Spherical Functions and Spherical Laplace Transform on Ordered Symmetric Spaces, (1997) 
  11. E.M. Opdam, Lectures on Dunkl Operators, (1998) 
  12. N.R. Wallach, Real Reductive Groups I, Vol. 132 (1988), Academic Press Zbl0666.22002MR929683

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.