The hypoelliptic Laplacian is a family of operators indexed by $b \in \mathbf{R}^*_+$, acting on the total space of the tangent bundle of a Riemannian manifold, that interpolates between the ordinary Laplacian as $b \to 0$ and the generator of the geodesic flow as $b \to +\infty$. These operators are not elliptic, they are not self-adjoint, they are hypoelliptic. One can think of the total space of the tangent bundle as the phase space of classical mechanics; so that the hypoelliptic Laplacian produces an interpolation between the geodesic flow and its quantisation. There is a dynamical counterpart, which is a natural interpolation between classical Brownian motion and the geodesic flow.
The hypoelliptic deformation preserves subtle invariants of the Laplacian. In the case of locally symmetric spaces (which are defined via Lie groups), the deformation is essentially isospectral, and leads to geometric formulas for orbital integrals, a key ingredient in Selberg's trace formula.
In a first part of the talk, I will describe the geometric construction of the hypoelliptic Laplacian in the context of de Rham theory. In a second part, I will explain applications to the trace formula.