Higher genera for proper actions of Lie groups

Let G be a Lie group with finitely many connected components and let K be a maximal compact subgroup. We assume that G satisfies the rapid decay (RD) property and that G∕K has a nonpositive sectional curvature. As an example, we can take G to be a connected semisimple Lie group. Let M be a G-proper manifold with compact quotient M∕G. Building on work by Connes and Moscovici (1990) and Pflaum et al. (2015), we establish index formulae for the C∗-higher indices of a G-equivariant Dirac-type operator on M. We use these formulae to investigate geometric properties of suitably defined higher genera on M. In particular, we establish the G-homotopy invariance of the higher signatures of a G-proper manifold and the vanishing of the ˆA-genera of a G-spin G-proper manifold admitting a G-invariant metric of positive scalar curvature.