Description
Dirac-type operators and Howe dualities provide a uniform approach to decomposition problems and symmetry algebras in the orthogonal and symplectic settings. Quantizing these structures is subtle and in the orthogonal Dirac setting competing frameworks do not currently agree.
In this talk I focus on the symplectic (metaplectic) Howe duality in the first nontrivial case, rank one, and present an explicit and computable quantization, based on joint work with M. Brito. The main outcome is a clean quantum duality in which both sides are Drinfeld–Jimbo sl(2)-type quantum groups (with different deformation parameters), realized via Hayashi’s deformed Weyl algebra. I will outline the resulting quantum analogues of the classical realization, Fischer-type structure, monogenics, and first-order symmetries, and conclude with the main obstacles to extending the construction beyond rank one.