Homogeneous 2-nondegenerate CR manifolds of hypersurface type, maximally symmetric relative to their modified CR symbols

We will cover applications of the relationship between homogeneous 2-nondegenerate CR manifolds of hypersurface type and their modified CR symbols, a local invariant of their structure recently defined (in joint work with Igor Zelenko) that also encodes the basic local invariants sometimes referred to as their generalized Levi forms. From this relationship we will derive algebraic criteria for a set of generalized Levi forms to admit a homogeneous model, and, in particular, obtain obstructions to homogeneity expressed in terms of the local invariants at a point. From these criteria, we obtain the classification up to local equivalence of 2-nondegenerate real hypersurfaces in complex 4-space that are locally equivalent to homogeneous CR manifolds whose symmetry groups have maximal dimension relative to their modified CR symbols. In total, there are 9 CR structures in this classification. In higher dimensions, by applying a Tanaka-theoretic algebraic prolongation to special reductions of the modified symbols we obtain estimates on the homogeneous models' symmetry group dimensions.