Highest Weight Modules for Affine and Loop Superalgebras of \(\mathfrak {osp}_{1|2}(\mathbb C)\)

This paper is about the highest weight module theory for affine superalgebra \(\widetilde{\mathfrak g}\) $\tilde{\mathfrak{g}}$ of \({\mathfrak g}={\mathfrak {osp}_{1|2}(\mathbb C)}\) $\mathfrak{g}={\mathfrak{osp}}_{1|2}(\mathbb{C})$ and loop superalgebra \({\mathfrak g}{\otimes }{\mathbb {C}}[t,t^{-1}]\) $\mathfrak{g}\otimes \mathbb{C}[t,t^{-1}]$ . Among the main results, we obtain (i) a necessary and sufficient condition for Verma type \(\ell \) $\ell$ -highest weight \(\widetilde{\mathfrak g}\) $\tilde{\mathfrak{g}}$ -modules to be irreducible; (ii) a free field(-like) realization of all irreducible \(\ell \) $\ell$ -highest weight \(\widetilde{\mathfrak g}\) $\tilde{\mathfrak{g}}$ -modules; (iii) a character formula for all irreducible \(\ell \) $\ell$ -highest weight \(\widetilde{\mathfrak g}\) $\tilde{\mathfrak{g}}$ -modules with finite dimensional weight spaces. We also obtain three similar results for highest weight \({\mathfrak g}{\otimes }{\mathbb {C}}[t,t^{-1}]\) $\mathfrak{g}\otimes \mathbb{C}[t,t^{-1}]$ -modules.