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Hypoelliptic Laplacian and Orbital Integrals (AM-177) by Jean-Michel Bismut

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Chapter Seven

An application to local index theory

The purpose of this chapter is to verify the compatibility of our formula in Theorem 6.1.1 for the orbital integrals of heat kernels to the index formula of Atiyah-Singer [AS68a, AS68b], to the fixed point formulas of Atiyah-Bott [ABo67, ABo68], and to the index formula for orbifolds of Kawasaki [Ka79]. Recall that the McKean-Singer formula [McKS67] expresses the index of a Dirac operator over a compact manifold Z as the supertrace of a heat kernel. If Z is the quotient of X by a cocompact torsion free group, this supertrace can be evaluated explicitly by the formulas we gave in chapter 6. Here we will directly check these formulas to be compatible with the index formulas.

This chapter is organized ...

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