Let be a Lie group with finitely many connected components and let be a maximal compact subgroup. We assume that satisfies the rapid decay (RD) property and that has a nonpositive sectional curvature. As an example, we can take to be a connected semisimple Lie group. Let be a -proper manifold with compact quotient . Building on work by Connes and Moscovici (1990) and Pflaum et al. (2015), we establish index formulae for the -higher indices of a -equivariant Dirac-type operator on . We use these formulae to investigate geometric properties of suitably defined higher genera on . In particular, we establish the -homotopy invariance of the higher signatures of a -proper manifold and the vanishing of the -genera of a -spin -proper manifold admitting a -invariant metric of positive scalar curvature.
"Higher genera for proper actions of Lie groups." Ann. K-Theory 4 (3) 473 - 504, 2019. https://doi.org/10.2140/akt.2019.4.473