Given a Riemannian manifold \(M\), a hypersurface \(N\) is called isoparametric if it is the regular level set of a so-called isoparametric function. When \(M\) is the round sphere \(S^n\), this is equivalent to \(N\) having constant principal curvatures, hence homogeneous hypersurfaces in \(S^n\) are isoparametric. Isoparametric hypersurfaces in \(S^n\) with \(g\) distinct principal curvatures are also characterized as regular level sets of the restriction to \(S^n\) of Cartan-Münzner polynomials of degree \(g\).
The authors expect the following result for the case \(g=4\): Every Cartan-Münzner polynomials is the squared norm of a moment map. The main statement of the paper works toward this result in the special case of isoparametric hypersurfaces of \(S^n\) which are suitable orbits of the isotropy representation of a compact irreducible symmetric space \(G/K\) of rank 2.
The isotropy representation of \(G/K\) is the adjoint action of \(K\) on \(\mathfrak{p}\), where \(\mathfrak{k}\oplus \mathfrak{p}\) is the Cartan decomposition of the Lie algebra of \(G\). \(\mathfrak{p}\) is endowed with a canonical (up to sign) symplectic form, and the isotropy action is Hamiltonian with canonical moment map. The main statement of the paper is that there exists an invariant norm on \(\mathfrak{k}^*\) such that the squared-norm of the moment map is a Cartan-Münzner polynomial of degree 4.