We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras \((\mathfrak {sp}_{2n},\mathfrak {sl}_2)\) in the case when \(n=1\) . Our results yield commuting representations of the pair of Drinfeld–Jimbo quantum groups \((\mathcal {U}_{q^2}(\mathfrak {sl}_2),\mathcal {U}_{q}(\mathfrak {sl}_2))\) 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.
