Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I: Rank-one actions on the centre

Véronique Fischer, Fulvio Ricci, Oksana Yakimova

Research output: Contribution to journalArticle

11 Citations (Scopus)

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 languageEnglish
Pages (from-to)221-255
Number of pages35
JournalMathematische Zeitschrift
Volume271
Issue number1-2
Early online date22 Mar 2011
DOIs
Publication statusPublished - 1 Jun 2012

Keywords

  • Gelfand pairs
  • Invariants
  • Schwartz functions
  • Spherical transform

ASJC Scopus subject areas

  • Mathematics(all)

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