Algebra and quantum geometry of multifrequency resonance

The algebra of symmetries of a quantum resonance oscillator in the case of three or more frequencies is described using a finite (minimal) basis of generators and polynomial relations. For this algebra, we construct quantum leaves with a complex structure (an analogue of classical symplectic leaves) and a quantum Kähler 2-form, a reproducing measure, and also the corresponding irreducible representations and coherent states.