The index of transversally elliptic operators for locally free actions

Jeffrey Stephen Fox and Peter Evarts Haskell

Vol. 164 (1994), No. 1, 41–85

DOI: 10.2140/pjm.1994.164.41

Abstract

Let a connected unimodular Lie group G act smoothly and locally freely on a closed manifold X. Assume that the isotropy groups of the action are torsion-free. Let K be the maximal compact subgroup of G. Let T be a G-invariant first order differential operator on X that is elliptic in directions transverse to the G-orbits. Using Kasparov products over C^∗G, we prove index formulas equating indices of elliptic operators on K∖X with linear combinations of multiplicities of G-representations in kernel(T) − kernel(T^∗).

Mathematical Subject Classification 2000

Primary: 58G12

Secondary: 19K35, 22E45, 46L89

Milestones

Received: 11 May 1992

Published: 1 May 1994