#### Vol. 4, No. 3, 2019

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Higher genera for proper actions of Lie groups

### Paolo Piazza and Hessel B. Posthuma

Vol. 4 (2019), No. 3, 473–504
##### 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 $\stackrel{̂}{A}$-genera of a $G$-spin $G$-proper manifold admitting a $G$-invariant metric of positive scalar curvature.

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Lie groups, proper actions, group cocycles, van Est isomorphism, cyclic cohomology, $K\mkern-2mu$-theory, index classes, higher indices, higher index formulae, higher signatures, $G$-homotopy invariance, higher genera, positive scalar curvature