Howe duality for the dual pair $\left(\text{SpO}(2n|1),\mathfrak{osp}(2|2)\right)$

Abstract

The goal of our work is to study the decomposition of the joint action of $\mathcal{G}=\text{SpO}(2n|1)$ and ${\mathfrak{g}}^{\prime }=\mathfrak{osp}(2|2)$ on the supersymmetric algebra $\text{S}=\text{S}({\mathbb{C}}^{2n|1}\otimes {\mathbb{C}}^{1|1})$. As proved by Merino and Salmasian, we have a one-to-one correspondence between irreducible representations of $\mathcal{G}$ and ${\mathfrak{g}}^{\prime }$ appearing as subrepresentations of $\text{S}$. In this paper, we obtained an explicit description of the highest weights and joint highest weight vectors for the representations of $\mathcal{G}$ and ${\mathfrak{g}}^{\prime }$ appearing in the duality.