The hypoelliptic Laplacian on a compact Lie group
Several years ago, the author [J. Am. Math. Soc. 18, No. 2, 379--476 (2005; Zbl 1065.35098)] introduced the hypoelliptic Laplacian into geometry, and its interplay with the Laplacian on the base manifold and the geodesic flow on its cotangent bundle. In [\textit{J.-M. Bismut} and \textit{G. Lebeau}, The hypoelliptic Laplacian and Ray-Singer metrics. Ann. Math. Stud. 167. Princeton, NJ: Princeton University Press (2008; Zbl 1156.58001)], the authors gave an analysis foundation of this theory and applications to index theory. In the paper under review, Bismut gives an application of his theory to the precise formula of the heat kernel on a compact Lie group \(G\) with Lie algebra \(\mathfrak{g}\). In this case, the cotangent bundle is simply \(G\times \mathfrak{g}\). The motivation is from the paper of \textit{M. F. Atiyah} [Circular symmetry and stationary-phase approximation. Colloq. Honneur L. Schwartz, Éc. Polytech. 1983, Vol. 1, Astérisque 131, 43--59 (1985; Zbl 0578.58039)], where Atiyah elaborated a remark by Witten on the relation of the index theorem of Atiyah-Singer on a compact spin manifold \(X\) to the Duistermaat-Heckman localization formula in equivariant cohomology over the loop space \(LX\) of \(X\). At the end of his paper, Atiyah also observed that a similar formal argument could be applied to the standard heat kernel on a compact Lie group. Note that the hypoelliptic Laplacian should be considered as a semi-classical version of the Witten Laplacian on loop spaces. Bismut found that the trace of the heat kernel does not change when we deform the Laplacian on the Lie group to the hypoelliptic Laplacian on \(G\times \mathfrak{g}\). This is the analogue of the McKean-Singer formula in local index theory. Then in the spirit of the proof of the local index theorem, we compute the limit when the parameter goes to \(0\) and to \(\infty\), and we get the final result. Bismut applied this idea to a reductive Lie group in the companion paper [C. R., Math., Acad. Sci. Paris 347, 1189--1195 (2009; Zbl 1183.58001)]. In this way, he proved the Selberg trace formula for locally symmetric spaces of non-compact type.
infinite-dimensional Lie groups and their Lie algebras
index theory and related fixed point theorems
scientific article; zbMATH DE number 5505654
scientific article