The Cartan-Helgason theorem for supersymmetric spaces: spherical weights for Kac-Moody superalgebras

Let (g, k) be a supersymmetric pair arising from a finite-dimensional, symmetrizable Kac-Moody superalgebra g. An important branching problem is to determine the finite-dimensional highest-weight g-modules which admit a k-coinvariant, and thus appear as functions in a corresponding supersymmetric space G/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 g is indecomposable.