Intertwining operators of the Heisenberg group (Q2769868)

It is well known that the Heisenberg group has three irreducible unitary representations, i.e., the Schrödinger, the Fock and the lattice representations. The author constructs an integral operator whose kernel is a function (called primitive vector) characterized by certain properties related to a polarization of the Fock representation, and shows that it is a known intertwining operator from the Schrödinger representation onto the Fock representation. He then constructs an integral operator whose kernel is the primitive vector for the lattice representation, and shows that it gives a known intertwining operator from the Schrödinger representation onto the lattice representation. Finally he constructs two integral operators whose kernels are difference primitive vectors for the Schrödinger representation, and shows that they give intertwining operators from the Fock representation onto the two difference realizations of the Schrödinger representation. As a consequence, the Jacobi identity for the theta series is equivalent to the commutativity of these intertwining operators.