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

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Introduction

The purpose of this book is to use the hypoelliptic Laplacian to evaluate semisimple orbital integrals, in a formalism that unifies index theory and the trace formula.

0.1 The trace formula as a Lefschetz formula

Let us explain how to think formally of such a unified treatment, while allowing ourselves a temporarily unbridled use of mathematical analogies. Let X be a compact Riemannian manifold, and let ΔX be the corresponding Laplace-Beltrami operator. For t > 0, consider the trace of the heat kernel Tr [exp (tΔX/2)]. If image is the Hilbert space of square-integrable functions on X, Tr [exp (tΔX/2)] is the trace of the ‘group element’ ...

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