Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup

In this paper, we obtain a full classification of reductive dual pairs in a, real or complex, Lie superalgebra $\mathfrak{s}\mathfrak{p}\mathfrak{o}(E)$ and Lie supergroup $\mathbf{\text{SpO}}(E)$. Moreover, by looking at the natural action of the orthosymplectic Lie supergroup $\mathbf{\text{SpO}}(E)$ on the Weyl-Clifford algebra $\mathbf{\text{WC}}(E)$, we prove that for a reductive dual pair $(\mathcal{G},{\mathcal{G}}^{\mathrm{\prime }})=((G,\mathfrak{g}),(G^{\mathrm{\prime }},{\mathfrak{g}}^{\mathrm{\prime }}))$ in $\mathbf{\text{SpO}}(E)$, the superalgebra $\mathbf{\text{WC}}(E{)}^{\mathcal{G}}$ consisting of $\mathcal{G}$-invariant elements in $\mathbf{\text{WC}}(E)$ is generated by the Lie superalgebra ${\mathfrak{g}}^{\mathrm{\prime }}$. We obtain a full classification of reductive dual pairs in the (real or complex) Lie superalgebra $\mathfrak{s}\mathfrak{p}\mathfrak{o}(\mathrm{E})$ and the Lie supergroup $\mathbf{\text{SpO}}(\mathrm{E})$. Using this classification we prove that for a reductive dual pair $(\mathcal{G},{\mathcal{G}}^{\mathrm{\prime }})=((\mathrm{G},\mathfrak{g}),({\mathrm{G}}^{\mathrm{\prime }},{\mathfrak{g}}^{\mathrm{\prime }}))$ in $\mathbf{\text{SpO}}(\mathrm{E})$, the superalgebra $\mathbf{\text{WC}}(\mathrm{E}{)}^{\mathcal{G}}$ consisting of $\mathcal{G}$-invariant elements in the Weyl-Clifford algebra $\mathbf{\text{WC}}(\mathrm{E})$, equipped with the natural action of the orthosymplectic Lie supergroup $\mathbf{\text{SpO}}(\mathrm{E})$, is generated by the Lie superalgebra ${\mathfrak{g}}^{\mathrm{\prime }}$. As an application, we prove that Howe duality holds for the dual pairs $(\mathbf{\text{SpO}}(2n\mathrm{\mid }1),\mathbf{\text{OSp}}(2k\mathrm{\mid }2l))\subseteq \mathbf{\text{SpO}}({\mathbb{C}}^{2k\mathrm{\mid }2l}\otimes {\mathbb{C}}^{2n\mathrm{\mid }1})$.