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 via a rank one reduction and an understanding of reflections in singular roots, which generalize odd reflections in the theory of Kac-Moody superalgebras. An explicit presentation of spherical weights is provided for every pair when $\mathfrak{ g}$ is indecomposable.
Categories: math.RT math-ph math.MP