Hypoelliptic Laplacian and orbital integrals

The hypoelliptic Laplacian is a natural family of second order operators acting on the total space of the (co)tangent bundle of a smooth compact manifold, which interpolates between the classical Hodge Laplacian (in de Rham or Dolbeault theory) and the geodesic flow. It is essentially a weighted sum of the harmonic oscillator along the fibre and of the vector field generating the geodesic flow. This hypoelliptic deformation comes itself from a deformation of the associatd Hodge theory, and of the corresponding Dirac operator. The analytic properties of the hypoelliptic Laplacian have been established by Lebeau and ourselves.   If  G  is a reductive group with Lie algebra  g, we have applied this method to the explicit evaluation of semisimple orbital integrals.  If  X = G/K is the associated symmetric space, the hypoelliptic deformation of the Casimir acts on X \times g.  The orbital integrals are shown to be independent of the deformation parameter. Localization on closed geodesics leads to an explicit formula for the orbital ingegrals. In this lecture, I will review the above constructions and results.