Weyl–Schrödinger Representations of Heisenberg Groups in Infinite Dimensions
Weyl–Schrödinger Representations of Heisenberg Groups in Infinite Dimensions
Abstract We investigate the group $${\mathcal {H}}_{\mathbb {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>H</mml:mi><mml:mi>C</mml:mi></mml:msub></mml:math> of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space H . Irreducible representations of the Weyl–Schrödinger type on the space $$L^2_\chi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>L</mml:mi><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:math> of quadratically integrable $${\mathbb {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> -valued functions are described. Integrability is …