Maximal dimension of groups of symmetries of homogeneous 2-nondegenerate CR structures of hypersurface type with a 1-dimensional Levi kernel

TL;DR

This paper establishes the maximum possible symmetry group dimension for certain homogeneous CR structures with a 1-dimensional Levi kernel, confirming a conjecture relating symmetry and Levi nondegeneracy.

Contribution

It proves the exact upper bound for symmetry group dimensions of these CR structures and extends the result to less restrictive classes, supporting Beloshapka's conjecture.

Findings

Maximum symmetry group dimension is n^2+7 for these structures.

The result holds for both homogeneous and more general classes.

Supports the conjecture that maximal symmetry implies Levi nondegeneracy.

Abstract

We prove that for every $n\geq 3$ the sharp upper bound for the dimension of the symmetry groups of homogeneous, 2-nondegenerate, $(2n+1)$-dimensional CR manifolds of hypersurface type with a $1$-dimensional Levi kernel is equal to $n^2+7$, and simultaneously establish the same result for a more general class of structures characterized by weakening the homogeneity condition. This supports Beloshapka's conjecture stating that hypersurface models with a maximal finite dimensional group of symmetries for a given dimension of the underlying manifold are Levi nondegenerate.