Deformation and $K$-theoretic index formulae on boundary groupoids

Motivated by investigating K -theoretic index formulae for boundary groupoids of the form {\mathcal{H}}= M_{0} \times M_{0} \sqcup G \times M_{1} \times M_{1} \rightrightarrows M = M_{0} \sqcup M_{1}, where G is an exponential Lie group, we introduce the notion of a deformation from the pair groupoid , which makes sense for general Lie groupoids. Once there exists a deformation from the pair groupoid for a (general) Lie groupoid {\mathcal{G}}\rightrightarrows M , we are able to construct explicitly a deformation index map relating the analytic index on {\mathcal{G}} and the index on the pair groupoid M \times M , which in turn enables us to establish index formulae for (fully) elliptic (pseudo)differential operators on {\mathcal{H}} by applying the numerical index formula of M. J. Pflaum, H. Posthuma, and X. Tang. In particular, we find that the index is given by the Atiyah–Singer integral but does not involve any \eta -term in the higher codimensional cases. These results recover and generalize our previous results for renormalizable boundary groupoids via renormalized traces.