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

Abstract

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.