The hypoelliptic Laplacian

If $X$ is a Riemannian manifold, the Laplacian is a second order elliptic operator on $X$. The hypoelliptic Laplacian $L_{b}\vert_{b>0}$ is a family of operators acting on the total space $\mathcal{X}$ of the tangent bundle $TX$ (or of a larger vector bundle), that is supposed to interpolate between the elliptic Laplacian (when $b\to 0$) and the geodesic flow (when $b\to +\infty $). Up to lower order terms, $L_{b}$ is a weighted sum of the harmonic oscillator along the fibre $TX$ and of the generator of the geodesic flow. Every geometrically defined Laplacian, like the Hodge Laplacian in de Rham theory or in Dolbeault theory, has a natural hypoelliptic deformation.
In the talk, I will explain applications of the hypoelliptic deformation to the evaluation of orbital integrals, and also to the proof of a Riemann–Roch–Grothendieck theorem in Bott–Chern cohomology.