The Paley-Wiener Theorem and Limits of Symmetric Spaces

TL;DR

This paper extends the Paley-Wiener theorem to infinite-dimensional symmetric spaces by introducing the concept of propagation and analyzing direct limit spaces, building on previous work on invariant operators.

Contribution

It introduces a framework for applying the Paley-Wiener theorem to infinite-dimensional symmetric spaces via propagation and direct limits.

Findings

Extended Paley-Wiener theorem to infinite-dimensional spaces

Defined propagation for symmetric spaces

Analyzed invariant differential operators in this context

Abstract

We extend the Paley--Wiener theorem for riemannian symmetric spaces to an important class of infinite dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.