Electronic Research Announcements in Mathematical Sciences (ERA-MS)

The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations

Pages: 138 - 154, January 2010      doi:10.3934/era.2010.17.138

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Jochen Brüning - Institut für Mathematik, Humboldt Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany (email)
Franz W. Kamber - Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, United States (email)
Ken Richardson - Department of Mathematics, Texas Christian University, Fort Worth, Texas 76129, United States (email)

Abstract: In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications is an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

Keywords:  Equivariant, index, transversally elliptic, eta invariant, stratification, foliation.
Mathematics Subject Classification:  58J20, 53C12, 58J28, 57S15, 54H15.

Received: August 2010;      Revised: October 2010;      Available Online: December 2010.