Weyl–Schrödinger Representations of Heisenberg Groups in Infinite Dimensions

Abstract

AbstractWe investigate the group $${\mathcal {H}}_{\mathbb {C}}$$HC of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space H. Irreducible representations of the Weyl–Schrödinger type on the space $$L^2_\chi $$Lχ2 of quadratically integrable $${\mathbb {C}}$$C-valued functions are described. Integrability is understood with respect to the projective limit $$\chi =\varprojlim \chi _i$$χ=lim←χi of probability Haar measures $$\chi _i$$χi defined on groups of unitary $$i\times i$$i×i-matrices U(i). The measure $$\chi $$χ is invariant under the infinite-dimensional group $$U(\infty )=\bigcup U(i)$$U(∞)=⋃U(i) and satisfies the abstract Kolmogorov consistency conditions. The space $$L^2_\chi $$Lχ2 is generated by Schur polynomials on Paley–Wiener maps. The Fourier-image of $$L^2_\chi $$Lχ2 coincides with the Hardy space $${H}^2_\beta $$Hβ2 of Hilbert–Schmidt analytic functions on H generated by the correspondingly weighted Fock space $$\varGamma _\beta (H)$$Γβ(H). An application to heat equation over $${\mathcal {H}}_{\mathbb {C}}$$HC is considered.