Higher order Levi forms on homogeneous CR manifolds (Q2231144)

Let \((M, T^{\ 0,1})\) be an abstract CR manifold. The Levi order at \(x \in M\) is defined as the smallest \(q\) for which, given any nonzero germ \(\overline Z \in {T_x}^{0,1},\) on can find \(p \le q,\) \(Z_1, \dots, Z_p \in {T_x}^{1,0}\) such that \[ [Z_1,[Z_2,\dots,[Z_p, \overline Z ]]] \not \in {T_x}^{1,0} \oplus {T_x}^{0,1} . \] (In case of Levi nondegeneracy at \(x\), the Levi order at \(x\) is \(1\).) The Levi order of a locally homogeneous CR manifold is defined in a similar way. See Definition 1.4. The interested reader could also consult the notions of finitely nondegeneracy and Levi number introduced in [\textit{M. S. Baouendi} et al., Real submanifolds in complex space and their mappings. Princeton, NJ: Princeton University Press (1999; Zbl 0944.32040)]. Theorems 2.11 and 2.17 are results regarding the bounds of the Levi order for some classes of homogeneous CR manifolds, while Proposition 3.2 shows the existence of homogeneous CR manifolds of any Levi order.