The fourfold way in real analysis. An alternative to the metaplectic representation

Applications of Lie groups to the sciences; explicit representations (22E70) Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics (81S30) General mathematical topics and methods in quantum theory (81Q99) Spaces with indefinite inner product (Kre?n spaces, Pontryagin spaces, etc.) (46C20) Harmonic analysis in several variables (42B99) Lagrangian submanifolds; Maslov index (53D12) Other special orthogonal polynomials and functions (33C47) Analysis on other specific Lie groups (43A80) Research exposition (monographs, survey articles) pertaining to topological groups (22-02)