# Nilpotent gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

## TL;DR

This paper introduces a bootstrapping method to prove the Schwartz extension property of spherical transforms for a broad class of nilpotent Gelfand pairs, based on quotient pairs and Hadamard formulas.

## Contribution

It develops a recursive approach to establish Schwartz extension property for new nilpotent Gelfand pairs using known cases and quotient structures.

## Key findings

- Method successfully applied to various nilpotent Gelfand pairs
- Enables recursive proof of property (S) for complex pairs
- Extends understanding of spherical transforms in harmonic analysis

## Abstract

It has been shown 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({\Sigma}) of functions on the Gelfand spectrum {\Sigma} of L^1(N)^K which extend to Schwartz functions on Rd, once {\Sigma} is suitably embedded in Rd. 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.