$q$-deformed Howe duality for orthosymplectic Lie superalgebras

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