Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II. Taylor expansion on singular sets

TL;DR

This paper advances the understanding of the spherical transform on nilpotent Gelfand pairs by proving a generalized Hadamard lemma and analyzing the Taylor expansion on singular sets, supporting the conjecture of an isomorphism with Schwartz functions.

Contribution

It proves a generalized Hadamard lemma for K-invariant functions on N and advances the proof of the conjecture relating Schwartz functions and the Gelfand spectrum.

Findings

Established a generalized Hadamard lemma for K-invariant functions.

Proved the Taylor expansion on singular sets for nilpotent Gelfand pairs.

Supported the conjecture of the spherical transform being an isomorphism.

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 properly 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 a direct sum of [n,n] and a K-invariant irreducible subspace.