On the Cartan–Helgason Theorem for Supersymmetric Pairs

Abstract Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair arising from a finite-dimensional, symmetrizable Kac–Moody superalgebra $\mathfrak{g}$. An important branching problem is to determine the finite-dimensional highest weight $\mathfrak{g}$-modules which admit a $\mathfrak{k}$-coinvariant, and thus appear as functions in a corresponding supersymmetric space $\mathcal{G}/\mathcal{K}$. This is the super-analogue of the Cartan–Helgason theorem. We solve this problem by generalizing the theory of odd reflections for superalgebras to that of singular reflections, and study their effect on representations. This allows us to reduce to the case of a rank one pair. An explicit presentation of spherical weights is provided for every pair when $\mathfrak{g}$ is indecomposable.