### Abstract

The spectrum of a Gelfand pair of the form (K× N,K), where N is a nilpotent group, can be embedded in a Euclidean space ℝ ^{d}. The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on ℝ ^{d} has been proved already when N is a Heisenberg group and in the case where N = N _{3,2} is the free two-step nilpotent Lie group with three generators, with K = SO _{3} (Astengo et al. in J Funct Anal 251:772-791, 2007; Astengo et al. in J Funct Anal 256:1565-1587, 2009; Fischer and Ricci in Ann Inst Fourier Gren 59:2143-2168, 2009). We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra n of N for all Gelfand pairs (K × N,K) in Vinberg's list (Vinberg in Trans Moscow Math Soc 64:47-80, 2003; Yakimova in Transform Groups 11:305-335, 2006).

Original language | English |
---|---|

Pages (from-to) | 221-255 |

Number of pages | 35 |

Journal | Mathematische Zeitschrift |

Volume | 271 |

Issue number | 1-2 |

Early online date | 22 Mar 2011 |

DOIs | |

Publication status | Published - 1 Jun 2012 |

### Fingerprint

### Keywords

- Gelfand pairs
- Invariants
- Schwartz functions
- Spherical transform

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

*271*(1-2), 221-255. https://doi.org/10.1007/s00209-011-0861-3

**Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I : Rank-one actions on the centre.** / Fischer, Véronique; Ricci, Fulvio; Yakimova, Oksana.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 271, no. 1-2, pp. 221-255. https://doi.org/10.1007/s00209-011-0861-3

}

TY - JOUR

T1 - Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I

T2 - Rank-one actions on the centre

AU - Fischer, Véronique

AU - Ricci, Fulvio

AU - Yakimova, Oksana

PY - 2012/6/1

Y1 - 2012/6/1

N2 - The spectrum of a Gelfand pair of the form (K× N,K), where N is a nilpotent group, can be embedded in a Euclidean space ℝ d. The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on ℝ d has been proved already when N is a Heisenberg group and in the case where N = N 3,2 is the free two-step nilpotent Lie group with three generators, with K = SO 3 (Astengo et al. in J Funct Anal 251:772-791, 2007; Astengo et al. in J Funct Anal 256:1565-1587, 2009; Fischer and Ricci in Ann Inst Fourier Gren 59:2143-2168, 2009). We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra n of N for all Gelfand pairs (K × N,K) in Vinberg's list (Vinberg in Trans Moscow Math Soc 64:47-80, 2003; Yakimova in Transform Groups 11:305-335, 2006).

AB - The spectrum of a Gelfand pair of the form (K× N,K), where N is a nilpotent group, can be embedded in a Euclidean space ℝ d. The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on ℝ d has been proved already when N is a Heisenberg group and in the case where N = N 3,2 is the free two-step nilpotent Lie group with three generators, with K = SO 3 (Astengo et al. in J Funct Anal 251:772-791, 2007; Astengo et al. in J Funct Anal 256:1565-1587, 2009; Fischer and Ricci in Ann Inst Fourier Gren 59:2143-2168, 2009). We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra n of N for all Gelfand pairs (K × N,K) in Vinberg's list (Vinberg in Trans Moscow Math Soc 64:47-80, 2003; Yakimova in Transform Groups 11:305-335, 2006).

KW - Gelfand pairs

KW - Invariants

KW - Schwartz functions

KW - Spherical transform

UR - http://www.scopus.com/inward/record.url?scp=84861011352&partnerID=8YFLogxK

U2 - 10.1007/s00209-011-0861-3

DO - 10.1007/s00209-011-0861-3

M3 - Article

VL - 271

SP - 221

EP - 255

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1-2

ER -