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

TL;DR

This paper extends the understanding of spherical transforms of Schwartz functions on nilpotent Gelfand pairs, showing that for pairs with spherical K-orbits in the center, the transforms can be identified with Schwartz functions on Euclidean space.

Contribution

It proves the identification of spherical transforms with Schwartz functions for all Gelfand pairs with spherical K-orbits in the center, generalizing previous specific cases.

Findings

Identification of spherical transforms with Schwartz functions for new classes of Gelfand pairs.

Construction of bases of K-invariant polynomials for all pairs in Vinberg's list.

Extension of known results from Heisenberg and free two-step nilpotent groups to broader cases.

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)