Higher genera for proper actions of Lie groups, Part 2: the case of manifolds with boundary

Abstract

Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C * -indices associated to elements [ϕ] ∈ H * diff (G) and to a generalized G-equivariant Dirac operator D on M with L 2invertible boundary operator D ∂ . We then establish a higher index formula for these C * -indices and use it in order to introduce higher genera for M , thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.

Higher orbital integrals, rho numbers and index theory

“…Using [45,Lemma 1.24] and the well known inequality | Tr(T )| ≤ T 1 for a smoothing operator on a smooth compact manifold 5 , we see that Tr S : A ∞ G (X, E) → C(G) so defined is a continuous map. (Even though [45,Lemma 1.24] uses a slightly different algebra on G instead of C(G), the proof showing that the trace norm A → ||A|| 1 , A ∈ Ψ −∞ (S) is continuous for the Fréchèt topology on Ψ −∞ (S) applies verbatim to show continuity in our case.) 5 with…”

“…Once again, we have here a projective tensor product (but note that Ψ −∞ (S) is nuclear). It can be proved, following [45,Prop. 1.7], that there are isomorphisms…”

“…Proof. Most of the arguments given in [45] use the closure under holomorphic calculus of certain algebras. This applies unchanged whether we use pseudodifferential operators based on the Harish-Chandra Schwartz algebra C(G) or the rapid decay algebra H ∞ L (G).…”

“…Applications were given once again to higher signatures and higher A-roof genera. This index theorem was extended to G-proper manifolds with boundary in the recent preprint [45]; this is a higher APS index theorem on G-proper manifolds for cyclic cocyles localized at the identity element.…”

“…We begin by stating the results on manifolds with boundary; these are obtained by combining the results of Hochs, Song and Tang, the use of (an extension of) Melrose's b-calculus to the present context developed in [45] and relative K-theory and relative cyclic cohomology techniques, as developed in [13,41,45].…”