CR Orbits on the complex flag manifolds of $G_2$

Abstract

We give an explicit description of the orbits of the split real form $G_{2(2)}$ on the complex flag manifolds of $G_2$ and study their induced invariant CR structures. In particular we study their Levi forms and show that all possible cases in the simple homogeneous setting occur, namely Levi nondegeneracy, $k$-nondegeneracy for $2\le k\le 3$, and holomorphic degeneracy. Moreover, following the classical approach to orbits on flag manifolds, we give equations for these homogeneous CR manifolds.

39 pages