Moment maps and isoparametric hypersurfaces of OT-FKM type (Q2037810)

This short paper complements the results of [the author, Math. Ann. 355, No. 3, 1067-1084 (2013; Zbl 1267.53057)] by giving a new description of the Cartan-Münzner polynomial appearing there in terms of moment maps. Isoparametric hypersurfaces of the standard Riemannian sphere \(S^{n+1}\) are hypersurfaces with constant principal curvatures. It is known that all such isoparametric hypersurfaces are given by a level set of the so-called Cartan-Münzner polynomial \(F(x)\) in \(\mathbb{R}^{n+2}\) restricted to the sphere. The paper focuses on the case of four distinct principal curvature. The main result is Theorem 5.2: setting \(n = 2l- 2\), it gives a formula for \(F(x)\) in terms of the moment map \(\mu\) -- in the sense of symplectic geometry -- of an action of the Spin group on the higher-dimensional space \(\mathbb{C}^{2l}\). Further, for isoparametric hypersurfaces of OT-FKM type (thus not homogeneous ones), in Theorem 6.1. the author shows that the image of the Gauss map of such a hypersurfaces lies inside the zero level set of a moment map suitably induced by \(\mu\).