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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