# Paley-Wiener isomorphism over infinite-dimensional unitary groups

## TL;DR

This paper extends the Paley-Wiener isomorphism to Hardy spaces over infinite-dimensional unitary groups, enabling analysis of group actions, generators, and applications to semigroups and representations.

## Contribution

It introduces an analog of the Paley-Wiener isomorphism for infinite-dimensional unitary groups and explores its applications to semigroups and irreducible representations.

## Key findings

- Established the Paley-Wiener isomorphism in infinite dimensions
- Analyzed shift and multiplicative groups on the Hardy space
- Connected the framework to Gauss-Weierstrass semigroups and Weyl-Schrödinger representations

## Abstract

An analog of the Paley-Wiener isomorphism for the Hardy space with an invariant measure over infinite-dimensional unitary groups is described. This allows us to investigate on such space the shift and multiplicative groups, as well as, their generators and intertwining operators. We show applications to the Gauss-Weierstrass semigroups and to the Weyl-Schr\"odinger irreducible representations of complexified infinite-dimensional Heisenberg groups.