Nilpotent Gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

Véronique Fischer, Fulvio Ricci, Oksana Yakimova

Research output: Contribution to journalArticlepeer-review

7 Citations (SciVal)
46 Downloads (Pure)


It has been shown [1,2,9,10] that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L 1(N) K commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N) K of K-invariant Schwartz functions on N and the space S(Σ) of functions on the Gelfand spectrum Σ of L 1(N) K which extend to Schwartz functions on R d, once Σ is suitably embedded in R d. We call this property (S). We present here a general bootstrapping method which allows to establish property (S) to new nilpotent pairs (N,K), once the same property is known for a class of quotient pairs of (N,K) and a K-invariant form of Hadamard formula holds on N. We finally show how our method can be recursively applied to prove property (S) for a significant class of nilpotent Gelfand pairs.

Original languageEnglish
Pages (from-to)1076-1128
Number of pages53
JournalJournal of Functional Analysis
Issue number4
Early online date28 Sept 2017
Publication statusPublished - 15 Feb 2018


  • Analysis on nilpotent Lie groups
  • Fourier and spectral multipliers
  • Gelfand pairs and spherical transform
  • Invariants
  • Primary
  • Secondary

ASJC Scopus subject areas

  • Analysis


Dive into the research topics of 'Nilpotent Gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs'. Together they form a unique fingerprint.

Cite this