The noncommutative infinitesimal equivariant index formula. II. (Q283248)

The paper extends the results of [the author, J. K-Theory 14, No. 1, 73--102 (2014; Zbl 1348.58015)] where the truncated infinitesemal equivariant Chern-Connes character was computed. If \(M\) is a compact, Riemannian, spin manifold, \(D\) the Dirac operator on the spinor bundle and \(G\) a compact connected Lie group acting on \(M\) by orientation preserving isometries which commutes with \(D\), the infinitesimal equivariant JLO cocycle is an entire cochain. The equivariant index is presented as a pairing of the equivariant infinitesimal Chern-Connes characters with the Chern character of an idempotent. The paper provides an explicit formula for the cocycle using the asymptotic pseudodifferential operators and shows the existence of the limit of infinitesimal equivariant Chern-Connes characters when the parameter \(t \to 0\). Similarly, first, truncated infinitesimal equivariant eta cochains are constructed and again, using similar methods, the truncated order is dropped and the regularity at zero of infinitesimal eta cochains is demonstrated. Finally, the results are extended to family case.