Moment maps and isoparametric hypersurfaces of OT-FKM type

Associated with a Clifford system on ℝ2l, a Spin(m + 1) action is induced on ℝ2l. An isoparametric hypersurface N in S2l−1 of OT-FKM (Ozeki, Takeuchi, Ferus, Karcher and Münzner) type is invariant under this action, and so is the Cartan-Münzner polynomial F(x). This action is extended to a Hamiltonian action on ℂ2l. We give a new description of F(x) by the moment map μ:ℂ2l→k*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu :{\mathbb{C}^{2l}} \to \mathfrak{t}*$$\end{document}, where k≅o(m+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{t}\cong \mathfrak{o}\left({m + 1} \right)$$\end{document} is the Lie algebra of Spin(m + 1). It also induces a Hamiltonian action on ℂP2l−1. We consider the Gauss map G of N into the complex hyperquadric ℚ2l−2(ℂ) ⊂ ℂP2l−1, and show that G(N) lies in the zero level set of the moment map restricted to ℚ2l−2(ℂ).