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

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.