Higher genera for proper actions of Lie groups

Abstract

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 $C^{\ast }$-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 $\hat{A}$-genera of a $G$-spin $G$-proper manifold admitting a $G$-invariant metric of positive scalar curvature.