Gelfand pairs on the Heisenberg group and Schwartz functions

TL;DR

This paper establishes a topological isomorphism between $K$-invariant Schwartz functions on the Heisenberg group and Schwartz functions on the Gelfand spectrum, advancing harmonic analysis on these structures.

Contribution

It proves that the Gelfand transform provides a topological isomorphism for $K$-invariant Schwartz functions on the Heisenberg group, extending the understanding of harmonic analysis in this context.

Findings

Gelfand transform is a topological isomorphism for $K$-invariant Schwartz functions.

The Gelfand spectrum is homeomorphic to a closed subset of $ ^s$.

This work generalizes harmonic analysis on the Heisenberg group with symmetry.

Abstract

Let $\Hn$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a compact group of automorphisms of $\Hn$ such that $(K\ltimes \Hn,K)$ is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of $K$-invariant Schwartz functions on $\Hn$ and the space of Schwartz function on a closed subset of $\R^s$ homeomorphic to the Gelfand spectrum of the Banach algebra of $K$-invariant integrable functions on $\Hn$.