Abstract

Let $G$ be a finitely connected Lie group and let $K$ be a maximal compact subgroup. Let $M$ be a cocompact $G$-proper manifold with boundary, endowed with a $G$-invariant metric which is of product type near the boundary. Under additional assumptions on $G$, for example that it satisfies the rapid decay condition and is such that $G∕K$ has nonpositive sectional curvature, we define higher Atiyah–Patodi–Singer $C^{\ast }$-indices associated to elements $\left[\phi \right]\in H_{diff}^{\ast }\left(G\right)$ and to a generalized $G$-equivariant Dirac operator $D$ on $M$ with $L^2$-invertible boundary operator $D_{\partial }$. We then establish a higher index formula for these $C^{\ast }$-indices and use it in order to introduce higher genera for $M$, thus generalizing to manifolds with boundary the results that we have established in Part I. Our results apply in particular to a semisimple Lie group $G$. We use crucially the pairing between suitable relative cyclic cohomology groups and relative $K$-theory groups.

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