Gelfand transforms of S O ( 3 ) -invariant Schwartz functions on the free group N 3 , 2

Véronique Fischer; Fulvio Ricci[1]

  • [1] Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7 56126 Pisa (Italy)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 6, page 2143-2168
  • ISSN: 0373-0956

Abstract

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The spectrum of a Gelfand pair ( K N , K ) , where N is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz K -invariant functions on N . We also show the converse in the case of the Gelfand pair ( S O ( 3 ) N 3 , 2 , S O ( 3 ) ) , where N 3 , 2 is the free two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.

How to cite

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Fischer, Véronique, and Ricci, Fulvio. "Gelfand transforms of $SO(3)$-invariant Schwartz functions on the free group $N_{3,2}$." Annales de l’institut Fourier 59.6 (2009): 2143-2168. <http://eudml.org/doc/10450>.

@article{Fischer2009,
abstract = {The spectrum of a Gelfand pair $(K\ltimesN, K)$, where $N$ is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz $K$-invariant functions on $N$. We also show the converse in the case of the Gelfand pair $(SO(3)\ltimesN_\{3,2\}, SO(3))$, where $N_\{3,2\}$ is the free two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.},
affiliation = {Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7 56126 Pisa (Italy)},
author = {Fischer, Véronique, Ricci, Fulvio},
journal = {Annales de l’institut Fourier},
keywords = {Gelfand pair; Schwartz space; nilpotent Lie group},
language = {eng},
number = {6},
pages = {2143-2168},
publisher = {Association des Annales de l’institut Fourier},
title = {Gelfand transforms of $SO(3)$-invariant Schwartz functions on the free group $N_\{3,2\}$},
url = {http://eudml.org/doc/10450},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Fischer, Véronique
AU - Ricci, Fulvio
TI - Gelfand transforms of $SO(3)$-invariant Schwartz functions on the free group $N_{3,2}$
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 6
SP - 2143
EP - 2168
AB - The spectrum of a Gelfand pair $(K\ltimesN, K)$, where $N$ is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz $K$-invariant functions on $N$. We also show the converse in the case of the Gelfand pair $(SO(3)\ltimesN_{3,2}, SO(3))$, where $N_{3,2}$ is the free two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.
LA - eng
KW - Gelfand pair; Schwartz space; nilpotent Lie group
UR - http://eudml.org/doc/10450
ER -

References

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  1. F. Astengo, B. Di Blasio, F. Ricci, Gelfand transforms of polyradial Schwartz functions on the Heisenberg group, J. Funct. Anal. 251 (2007), 772-791 Zbl1128.43009MR2356428
  2. F. Astengo, B. Di Blasio, F. Ricci, Gelfand pairs on the Heisenberg group and Schwartz functions, (2008) Zbl1167.43008MR2490230
  3. C. Benson, J. Jenkins, G. Ratcliff, On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc. 321 (1990), 85-116 Zbl0704.22006MR1000329
  4. C. Benson, J. Jenkins, G. Ratcliff, The spherical transform of a Schwartz function on the Heisenberg group, J. Funct. Anal. 154 (1998), 379-423 Zbl0914.22013MR1612717
  5. F. Ferrari R., The topology of the spectrum for Gelfand pairs on Lie groups, Bull. Un. Mat. It. 10 (2007), 569-579 Zbl1177.22004MR2351529
  6. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207 Zbl0312.35026MR494315
  7. G. B. Folland, E. M. Stein, Hardy spaces on homogeneous groups, 28 (1982), Princeton University Press, Princeton, N.J. Zbl0508.42025MR657581
  8. D. Geller, Fourier analysis on the Heisenberg group. I. Schwartz space, J. Funct. Anal. 36 (1980), 205-254 Zbl0433.43008MR569254
  9. D. Geller, Liouville’s theorem for homogeneous groups, Comm. Partial Differential Equations 8 (1983), 1665-1677 Zbl0538.35019MR729197
  10. R. Goodman, N. R. Wallach, Representations and invariants of the classical groups, 68 (1998), Cambridge University Press, Cambridge Zbl0948.22001MR1606831
  11. B. Helffer, J. Nourrigat, Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differential Equations 4 (1979), 899-958 Zbl0423.35040MR537467
  12. S. Helgason, Groups and geometric analysis, 113 (1984), Academic Press Inc., Orlando, FL Zbl0543.58001MR754767
  13. S. Helgason, The Radon transform, 5 (1999), Birkhäuser Boston Inc., Boston, MA Zbl0932.43011MR1723736
  14. A. Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math. 78 (1984), 253-266 Zbl0595.43007MR782662
  15. J. N. Mather, Differentiable invariants, Topology 16 (1977), 145-155 Zbl0376.58002MR436204
  16. G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63-68 Zbl0297.57015MR370643
  17. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, 43 (1993), Princeton University Press, Princeton, NJ Zbl0821.42001MR1232192

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