We will construct a family of irreducible representation of the Heisenberg Lie algebra parametrized by pairs of transverse complex Lagrangian subspaces of a complexified symplectic vector space. This family recovers the families constructed by Satake, Mumford, and Grossman-Daubechies. Moreover, it recovers the classical Schrodinger and Fock-Bargmann representations that are fundamental to quantum mechanics.
Part I: First we will discuss how representations of the Heisenberg group arise from translational symmetries on symplectic vector spaces. We will review Grossman's observation of how this action preserves J-holomorphicity, and generalize this idea to construct the operators of the representations of the Heisenberg Lie algebra. Next, we will motivate the use of complex Lagrangian subspaces as parameters. In addition to how pairs of transverse complex Lagrangian subspaces relate to compatible complex structures, we will describe how in the symplectic setting, Grassmannians of subspaces of various dimensions are homotopic to the strata in the complex Lagrangian Grassmannian. We will give a pictorial description of the parameter space and the reconstruction dictionary in the two dimensional setting.
Part II: First we will review the technical precautions in infinite dimensional representation theory, and confirm that the construction abides by these requirements. We will prove the irreducibility of the constructed representations from the uniqueness of the ground state vector. The failure of exponentiability will be noted. Next we will present some results in an ongoing joint work with G. Khan. We will briefly review the relationship between the Jacobi group and the Eisenstein series, and show how the ground state vector can be used to factor the action of the Jacobi group. Finally, we will provide examples of how the infinite dimensional Fubini-Study metric reproduces some classical invariant metrics.