Geometry of G2 orbits and isoparametric hypersurfaces

Abstract

We characterize the adjoint $G_2$ orbits in the Lie algebra $\mathfrak{g}$ of $G_2$ as fibered spaces over $S^6$ with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case $(g,m)=(6,2)$ with case $(3,2)$. The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations of $S^6$ and $G_2/SO\left(4\right)$. From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.