$q$-deformed Howe duality for orthosymplectic Lie superalgebras

TL;DR

This work constructs a $q$-analogue of Howe duality for orthosymplectic Lie superalgebras by pairing ${\mathcal{U}}_X(\epsilon)$ with $\imath$-quantum groups acting on a $q$-deformed superspace. Through explicit $q$-oscillator realizations $\mathcal{W}$ and $\mathcal{W}^2$, it yields commuting actions and a semisimple, multiplicity-free decomposition whose classical limit recovers the traditional $(\mathfrak{g},G)$-duality, including ($q$-deformed) symmetric and exterior algebra analogues for $\mathfrak{so}_{2n}$ and $\mathfrak{sp}_{2n}$. The paper develops two quantum symmetric pair settings, AI and AII, corresponding to $(\mathfrak{sl}_\ell,\mathfrak{so}_\ell)$ and $(\mathfrak{sl}_{2\ell},\mathfrak{sp}_{2\ell})$, and proves a joint action on tensor powers that yields a robust $(\mathcal{U}_X(\epsilon), {\bf U}^\imath(\mathfrak{k}))$-duality on $\mathscr{W}^{\otimes\ell}$. It also clarifies the classical limits, semisimplicity criteria, and polarization structures that underpin the duality, connecting to broader super duality frameworks. Overall, the results extend known $q$-duality phenomena to orthosymplectic settings and provide explicit, computable decompositions with potential crystal/ combinatorial interpretations.

Abstract

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