Moment maps and isoparametric hypersurfaces of OT-FKM type

Abstract

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 Munzner) type is invariant under this action, and so is the Cartan-Munzner 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 $$\mu :{\mathbb{C}^{2l}} \to \mathfrak{t}*$$ , where $$\mathfrak{t}\cong \mathfrak{o}\left({m + 1} \right)$$ 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).