## DG - MP - PDE Seminar: Jean-Michel Bismut (U. Paris-Sud, Orsay, France)

- Date: 09/27/2011
- Time: 15:30

University of British Columbia

"Orbital integrals and the hypoelliptic Laplacian"

Abstract:

If G is a reductive Lie group with Lie algebra g, orbital integrals are a key ingredient in Selberg's trace formula. I will explain how one can think of the evaluation of orbital integrals as the computation of a Lefschetz trace. Using in particular the Dirac operator of Kostant, the standard Casimir operator of X = G/K is deformed to a hypoelliptic operator L_b acting on the total space of a canonically flat vector bundle on X, that contains TX as a subbundle. The symbol of this hypoelliptic operator is exactly the one described in the previous talks. When descending the situation to a locally symmetric space, the spectrum of the original Casimir remains rigidly embedded in the spectrum of the hypoelliptic deformation. Making b -> +\infty gives an explicit evaluation of semisimple orbital integrals.

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