$q$-deformed Howe duality for orthosymplectic Lie superalgebras

We give a $q$-analogue of Howe duality associated to a pair $(\text{mf}g,G)$, where $\text{mf}g$ is an orthosymplectic Lie superalgebra and $G=O_{\mathrm{\ell }},S{p}_{2\mathrm{\ell }}$. We define explicitly {commuting actions} of a quantized enveloping algebra of $\text{mf}g$ and the $\mathrm{ı}$quantum group of {type AI and AII} on a $q$-deformed supersymmetric space, and describe its semisimple decomposition whose classical limit recovers the $(\text{mf}g,G)$-duality. As special cases, we obtain $q$-analogues of $(\text{mf}g,G)$-dualities on symmetric and exterior algebras for $\text{mf}g=\text{mf}{so}_{2n}$, $\text{mf}{sp}_{2n}$.