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 $\mathscr{G} = \text{SpO}(2n|1)$ and $\mathfrak{g}' = \mathfrak{osp}(2|2)$ 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 $\mathscr{G}$ and $\mathfrak{g}'$ 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 $\mathscr{G}$ and $\mathfrak{g}'$ appearing in the duality.