Weyl-Schr\"odinger representations of infinite-dimensional Heisenberg groups on symmetric Wiener spaces

We investigate the group ${\mathcal{H}}_{\mathbb{C}}$ of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space $H$. Irreducible representations of the Weyl–Schr{"o}dinger type on the space $L_{\chi }^2$ of quadratically integrable $\mathbb{C}$-valued functions are described. Integrability is understood with respect to the projective limit $\chi =\mathrm{}{\chi }_i$ of probability Haar measures ${\chi }_i$ defined on groups of unitary $i\times i$-matrices $U(i)$. The measure $\chi$ is invariant under the infinite-dimensional group $U(\mathrm{\infty })=\bigcup U(i)$ and satisfies the abstract Kolmogorov consistency conditions. The space $L_{\chi }^2$ is generated by Schur polynomials on Paley–Wiener maps. The Fourier-image of $L_{\chi }^2$ coincides with the Hardy space $H_{\beta }^2$ of Hilbert–Schmidt analytic functions on $H$ generated by the correspondingly weighted Fock space ${\mathrm{\Gamma }}_{\beta }(H)$. An application to heat equation over ${\mathcal{H}}_{\mathbb{C}}$ is considered.