  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
$L^2_\chi$ of quadratically integrable $\mathbb{C}$-valued functions are
described. Integrability is understood with respect to the projective limit
$\chi=\varprojlim\chi_i$ 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 $U(\infty)=\bigcup U(i)$ and satisfies the
abstract Kolmogorov consistency conditions. The space $L^2_\chi$ is generated
by Schur polynomials on Paley--Wiener maps. The Fourier-image of $L^2_\chi$
coincides with the Hardy space ${H}^2_\beta$ of Hilbert--Schmidt analytic
functions on $H$ generated by the correspondingly weighted Fock space
$\Gamma_\beta(H)$. An application to heat equation over
$\mathcal{H}_\mathbb{C}$ is considered.
