On the Quantum Metaplectic Howe Duality

We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras (sp₂₍, sl₂) in the case when n=1. Our results yield commuting representations of the pair of Drinfeld-Jimbo quantum groups (Uₐℂ (sl₂), Uₐ (sl₂) ) realized in a suitable algebra of q-differential operators acting on the space of symplectic polynomial spinors. We obtain q-analogues for the symplectic Dirac operator, the Fischer decomposition, the expression for the symplectic polynomial monogenics and for the projection operators onto the monogenics. We also discuss q-analogues of generalized symmetries of the q-symplectic Dirac operator raising the homogeneous polynomial degree.