Replacing the canonical pair q and p of the classical harmonic oscillator (HO) by the locally and symplectically equivalent pair angle ϕ and action variable I implies a qualitative change of the global topological structure of the associated phase spaces: the pair (q, p) is an element of a topologically trivial plane R 2 whereas the pair (ϕ, I > 0) ∈ S 1 × R+ is an element of a topologically nontrivial, infinitely connected, punctured plane R 2 −{0}, which has the orthochronous "Lorentz" group SO ↑ (1, 2) (or its two-fold covering, the symplectic group Sp(2, R)) as its "canonical" group. Due to its infinitely many covering groups the resulting ("symplectic") spectrum of the associated quantum Hamiltonian H = ω Î is given by { ω(n + b), n = 0, 1, . . . ; b ∈ (0.1], e.g. b = 1/s, s ∈ N and large}, in contrast to the (q, p) version, where the Hamiltonian has the "orthodox" spectrum { ω(n + 1/2)}. The deeper reason for the difference is that for the description of the periodic orbit {p = p(q)} one covering of S 1 suffices, whereas one generally needs many coverings for the time evolution ϕ(t). And this, in turn, can lead to a lowering of the zero-point energies.Several theoretical and possible experimental implications of the "symplectic" spectra of the HO are discussed: The potentially most important ones concern the vibrations of diatomic molecules in the infrared, e.g. those of molecular hydrogen H2. Those symplectic spectra of the HO may provide a simultaneous key to two outstanding astrophysical puzzles, namely the nature of dark (vacuum) energy and that of dark matter: To the former because the zero-point energy b ω of free electromagnetic wave oscillator modes can be extremely small > 0 (b ≈ exp (−35) for the measured dark energy density). And a key to the dark matter problem because the quantum zeropoint energies of the electronic Born-Oppenheimer potentials in which the two nuclei of H2 or the nuclei of other primordial diatomic molecules vibrate can be lower, too, and, therefore, may lead to spectrally detuned "dark" H2 molecules during the "Dark Ages" of the universe and forming WIMPs in the hypothesized sense! All results appear to be in surprisingly good agreement with the ΛCDM model of the universe.Besides laboratory experiments the search for 21-cm radio signals from the Dark Ages of the universe and other astrophysical observations can help to explore those hypothetical implications.