Classification and Double Commutant Property for Dual Pairs in an Orthosymplectic Lie Supergroup

Let \(\textrm{E}=\textrm{E}_{\bar{0}}\oplus \textrm{E}_{\bar{1}}\) $\text{E}={\text{E}}_{\overline{0}}\oplus {\text{E}}_{\overline{1}}$ be a real or complex \(\mathbb {Z}_2\) ${\mathbb{Z}}_2$ -graded vector space equipped with an even supersymmetric bilinear form that restricts to a symplectic form on \(\textrm{E}_{\bar{0}}\) ${\text{E}}_{\overline{0}}$ and an orthogonal form on \(\textrm{E}_{\bar{1}}\) ${\text{E}}_{\overline{1}}$ . We obtain a full classification of reductive dual pairs in the (real or complex) orthosymplectic Lie superalgebra \(\mathfrak {spo}\) $\mathfrak{spo}$ (E) and its associated Lie supergroup \({\textbf {SpO}}(\textrm{E})\) $\mathbf{SpO}(\text{E})$ . Similar to the purely even case, dual pairs are divided into two subclasses: Type I and Type II. The main difference with the purely even case occurs in the characterization of (super)hermitian forms on modules over division superalgebras. We then use this classification to prove that for a reductive dual pair \((\mathscr {G}\,, \mathscr {G}') = ((\textrm{G}\,, \mathfrak {g})\,, (\textrm{G}'\,, \mathfrak {g}'))\) $(\mathcal{G},{\mathcal{G}}^{\prime })=((\text{G},\mathfrak{g}),({\text{G}}^{\prime },{\mathfrak{g}}^{\prime }))$ in \({\textbf {SpO}}(\textrm{E})\) $\mathbf{SpO}(\text{E})$ , the superalgebra \({\textbf {WC}}(\textrm{E})^{\mathscr {G}}\) $\mathbf{WC}{(\text{E})}^{\mathcal{G}}$ that consists of \(\mathscr {G}\) $\mathcal{G}$ -invariant elements in the Weyl-Clifford algebra \({\textbf {WC}}(\textrm{E})\) $\mathbf{WC}(\text{E})$ , when it is equipped with the natural action of the orthosymplectic Lie supergroup \({\textbf {SpO}}(\textrm{E})\) $\mathbf{SpO}(\text{E})$ , is generated by the Lie superalgebra \(\mathfrak {g}'\) ${\mathfrak{g}}^{\prime }$ . As an application of the latter double commutant property, we prove that Howe duality holds for the dual pairs \(( {{\textbf {SpO}}}(2n|1)\,, {{\textbf {OSp}}}(2k|2l)) \subseteq {{\textbf {SpO}}}(\mathbb {C}^{2k|2l} \otimes \mathbb {C}^{2n|1})\) $(\mathbf{SpO}(2n|1),\mathbf{OSp}(2k|2l))\subseteq \mathbf{SpO}({\mathbb{C}}^{2k|2l}\otimes {\mathbb{C}}^{2n|1})$ .