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

Abstract

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