On homogeneous $k$-nondegenerate CR manifolds

In this talk, I will report on a method for building homogeneous manifolds with an invariant $k$-nondegenerate CR structure of hypersurface type. I will explain how to combine the Tanaka and Freeman filtrations of a CR manifold $(M,\mathcal D,\mathcal J)$ into a single filtration and construct an associated pointwise invariant $\mathfrak m_x=\mathfrak m_x^{-2}\oplus\mathfrak m_x^{-1}\oplus\mathfrak m_{x}^0\oplus\cdots\oplus\mathfrak m_x^{k-2}$, called the core at the point $x\in M$. The collection of all Levi forms $\mathcal L^{p+1}$ of higher degree induces operators $L^{p+1}$ on $\mathfrak m_x$ but, in sharp contrast with the nondegenerate case, the core does not possess any natural structure of a Lie algebra and the problem of constructing homogeneous $M=G/H$ with a given core is more involved.
The method is a generalization of Tanaka's construction of homogeneous models via prolongation of negatively-graded Lie algebras. We will recognize the $L^{p+1}$'s as defining components of Weisfeiler infinite-dimensional contact algebra $\mathfrak c$ and endow $\mathfrak c$ with a natural structure of a CR algebra. The germ of $M=G/H$ is then obtained as an appropriate CR subalgebra $\mathfrak g$ of $\mathfrak c$ that prolongs $\mathfrak m_x$. In the second part of the talk, I will consider applications in dimension $\dim M=7$. I will present the classification of the $2$-nondegenerate cores up to isomorphism and obtain seven not locally CR diffeomorphic homogeneous CR manifolds with given cores. Finally, there exists a $7$-dimensional $M=G/H$ corresponding to the unique $3$-nondegenerate core.