On the Quantum Metaplectic Howe Duality

We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras (sp₂ₙ,sl₂) in the case when $n=1$. Our results yield commuting representations of the pair of Drinfeld-Jimbo quantum groups ( Uq²(sl₂), Uq(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.