# Weyl-Schr\"odinger representations of Heisenberg groups in infinite dimensions

## TL;DR

This paper explores complexified Heisenberg groups in infinite dimensions, describing their Weyl--Schr"odinger representations on a specialized L^2 space, and applies these findings to heat equations on the group.

## Contribution

It introduces a new framework for representing infinite-dimensional Heisenberg groups using invariant measures and Schur polynomials, linking to Fock space and analytic functions.

## Key findings

- Representation of H_C on L^2 space described
- Connection established between L^2 space and Fock space
- Applications to heat equations on the group

## Abstract

A complexified Heisenberg matrix group $\mathrm{H}_\mathbb{C}$ with entries from an infinite-dimensional Hilbert space $H$ is investigated. The Weyl--Schr\"odinger type irreducible representations of $\mathrm{H}_\mathbb{C}$ on the space $L^2_\chi$ of square-integrable scalar functions is described. The integrability is understood under the invariant probability measure $\chi$ which satisfies an abstract Kolmogorov consistency conditions over the infinite-dimensional unitary group $U(\infty)$ irreducible acted on $H$. The space $L^2_\chi$ is generated by Schur polynomials in variables on Paley--Wiener maps over $U(\infty)$. Therewith, the Fourier-image of $L^2_\chi$ coincides with a space of Hilbert--Schmidt entire analytic functions on $H$ generated by suitable Fock space. Applications to linear and nonlinear heat equations over the group $\mathrm{H}_\mathbb{C}$ are considered.