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

Abstract

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

The first version of the paper has been revised substantially

References in corpus (1)

• Dual pairs in the Pin-group and duality for the corresponding spinorial representation