Weyl-Schrödinger 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{ö}dinger type on the space $L^2_χ$ of quadratically integrable $\mathbb{C}$-valued functions are described. Integrability is understood with respect to the projective limit $χ=\varprojlimχ_i$ of probability Haar measures $χ_i$ defined on groups of unitary $i\times i$-matrices $U(i)$. The measure $χ$ is invariant under the infinite-dimensional group $U(\infty)=\bigcup U(i)$ and satisfies the abstract Kolmogorov consistency conditions. The space $L^2_χ$ is generated by Schur polynomials on Paley--Wiener maps. The Fourier-image of $L^2_χ$ coincides with the Hardy space ${H}^2_β$ of Hilbert--Schmidt analytic functions on $H$ generated by the correspondingly weighted Fock space $Γ_β(H)$. An application to heat equation over $\mathcal{H}_\mathbb{C}$ is considered.