An equivariant Atiyah-Patodi-Singer index theorem for proper actions II: the $K$-theoretic index

Abstract

Consider a proper, isometric action by a unimodular locally compact group G on a Riemannian manifold M with boundary, such that M/G is compact. Then an equivariant Dirac-type operator D on M under a suitable boundary condition has an equivariant index index G (D) in the K-theory of the reduced group C * -algebra C * r G of G. This is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. In part I of this series, a numerical index index g (D) was defined for an element g ∈ G, in terms of a parametrix of D and a trace associated to g. An Atiyah-Patodi-Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions,for a trace τ g defined by the orbital integral over the conjugacy class of g. This implies that the index theorem from part I yields information about the K-theoretic index index G (D). It also shows that index g (D) is a homotopy-invariant quantity.

Higher orbital integrals, rho numbers and index theory

“…In this section we want to tackle the easiest case of a delocalized APS index theorem on G-proper manifolds with boundary, that of delocalized traces, that is 0-degree delocalized cyclic cocycles. This result was already discussed in the work of Hochs-Wang-Wang [20]. Our treatment is different, centred around the interplay between absolute and relative cyclic cohomology and b-calculus; moreover our treatment allows us to get sharper results compared to [20] in the case of a connected linear reductive Lie group 3 G. More precisely, in Theorem 1.10 we only assume that g is a semisimple element of G to obtain the index formula (1.11), while in [20, Theorem 2.1], the authors require that G/Z g is compact.…”

“…In fact, one can show that η g (D) is well defined on a cocompact G-proper manifold even if D does not arise as a boundary operator. This is the content of the next theorems, partially discussed also in [19,20]. We assume as always when short time limits of the heat kernels are involved, that the metric is slice-compatible.…”

“…First we give an improvement, through the above techniques, of a result already proved in [20]. Let G be a connected, linear, reductive Lie group.…”

Primary and secondary invariants of Dirac operators on $G$-proper manifolds

“…However, there are other approaches to the C * -index class. Indeed, in the recent preprint [26] a coarse approach is explained. It should not be difficult to show that the two index classes, the one defined here and the one defined in [26] are in fact the same.…”

“…Indeed, in the recent preprint [26] a coarse approach is explained. It should not be difficult to show that the two index classes, the one defined here and the one defined in [26] are in fact the same. (The proof should proceed as for Galois coverings, where the analogous result is established in [49,Proposition 2.4].…”