Hypoelliptic Laplacian and Hodge theories

The hypoelliptic Laplacian is a family of operators acting on the total space of the tangent bundle (or a larger bundle) of a manifold. It is supposed to interpolate in the proper sense between the classical elliptic Hodge Laplacian and the geodesic flow. It is not self-adjoint. In certain cases, the full spectrum of the original Laplacian remains rigidly embedded in the spectrum of the deformation.

I will explain how such operators arise naturally in the evaluation of semisimple orbital integrals for real reductive groups. Chern-Weil theory will appear as another sort of degenerate Hodge theory. Finally, I will describe the role of the hypoelliptic Laplacian in complex geometry.