Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II: Taylor expansions on singular sets

Véronique Fischer, Fulvio Ricci, Oksana Yakimova

Research output: Chapter in Book/Report/Conference proceedingChapter

2 Citations (Scopus)

Abstract

This paper is a continuation of [8] in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N, K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum ΣD, appropriately embedded in a Euclidean space. We prove a result, of independent interest for the representation-theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's condition. This means that the Lie algebra n of N (which is step 2) decomposes as v ⊕ [n,n] with v irreducible under K.

Original languageEnglish
Title of host publicationLie Groups
Subtitle of host publicationStructure, Actions, and Representations: In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday
EditorsA. Huckleberry, I. Penkov, G. Zuckerman
Place of PublicationNew York, U. S. A.
PublisherBirkhauser Boston
Pages81-112
Number of pages32
ISBN (Electronic)9781461471936
ISBN (Print)9781461471929
DOIs
Publication statusPublished - 1 Jan 2013

Publication series

NameProgress in Mathematics
Volume306

Keywords

  • Gelfand pairs
  • Invariants
  • Schwartz functions
  • Spherical transform

ASJC Scopus subject areas

  • Mathematics(all)

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    Fischer, V., Ricci, F., & Yakimova, O. (2013). Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II: Taylor expansions on singular sets. In A. Huckleberry, I. Penkov, & G. Zuckerman (Eds.), Lie Groups: Structure, Actions, and Representations: In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday (pp. 81-112). (Progress in Mathematics; Vol. 306). Birkhauser Boston. https://doi.org/10.1007/978-1-4614-7193-6_5