Adiabatic groupoid and secondary invariants in K-theory

In this paper we define new K-theoretic secondary invariants attached to a Lie groupoid G. The receptacle for these invariants is the K-theory of C r⁎ (G ad∘ ) (where G ad∘ is the adiabatic deformation G restricted to the interval [0,1)). Our construction directly generalises the cases treated in [29,30], in the setting of the Coarse Geometry, to more involved geometrical situations, such as foliations. Moreover we tackle the problem of producing a wrong-way functoriality between adiabatic deformation groupoid K-groups associated to transverse maps. This extends the construction of the lower shriek map in [6]. Furthermore we prove a Lie groupoid version of the Delocalized APS Index Theorem of Piazza and Schick. Finally we give a product formula for secondary invariants.