| Description |
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.
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