Nilpotent Gelfand Pairs and Spherical Transforms of Schwartz Functions I. Rank-One Actions on the Centre

2010/02/18 by Véronique Fischer, Fulvio Ricci, Fischer, Veronique +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1002.3630

Abstract

The spectrum of a Gelfand pair of the form (K lx N, K), where N is a nilpotent group, can be embedded in a Euclidean space Rd . The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on Rd has been proved already when N is a Heisenberg group and in the case where N = N3,2 is the free two-step nilpotent Lie group with three generators, with K = SO3 [2, 3, 11]. 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 lx N, K) in Vinberg's list [27, 30]. (The references numbers refers to the bibliography at the end of the article)

Cited by

• A novel approach through spherical functions in the characterization of invariant functions

Related

• Nilpotent gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs
• Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II. Taylor expansion on singular sets
• Nilpotent Gelfand pairs and spherical transforms of Schwartz functions III. Isomorphisms between Schwartz spaces under Vinberg's condition
• Gelfand transforms of SO(3)-invariant Schwartz functions on the free group N_3,2
• On the Schwartz correspondence for Gelfand pairs of polynomial growth