Moment maps and Isoparametric hypersurfaces in spheres --- Hermitian cases

TL;DR

This paper explores the connection between isoparametric hypersurfaces in spheres derived from Hermitian symmetric spaces and moment maps of Hamiltonian actions, revealing a new geometric interpretation of Cartan-Münzner polynomials.

Contribution

It demonstrates that Cartan-Münzner polynomials for certain isoparametric hypersurfaces can be expressed as squared norms of moment maps, linking symmetric space theory with Hamiltonian geometry.

Findings

Cartan-Münzner polynomials are squared norms of moment maps.

Isoparametric hypersurfaces relate to Hermitian symmetric spaces.

The proof uses symmetric space structure theory.

Abstract

We are studying a relationship between isoparametric hypersurfaces in spheres with four distinct principal curvatures and the moment maps of certain Hamiltonian actions. In this paper, we consider the isoparametric hypersurfaces obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. We prove that the Cartan-M\"unzner polynomials of these hypersurfaces can be written as squared-norms of the moment maps for some Hamiltonian actions. The proof is based on the structure theory of symmetric spaces.