Moment maps and isoparametric hypersurfaces in spheres — Grassmannian cases

Abstract

We expect that every Cartan–Münzner polynomial of degree four can be described as a squared-norm of a moment map for a Hamiltonian action. Our expectation is known to be true for Hermitian cases, that is, those obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. In this paper, we prove that our expectation is true for the Cartan–Münzner polynomials obtained from the isotropy representations of Grassmannian manifolds of rank two over $R$, $C$ or $H$. The quaternion cases are the first non-Hermitian examples that our expectation is verified.