Delocalized eta invariants of the signature operator on G-proper manifolds

Abstract: Let $G$ be a connected, linear real reductive group and let $X$ be a cocompact $G$-proper manifold without boundary. We define delocalized eta invariants associated to a $L^2$-invertible perturbed Dirac operator $D_X+A$ with $A$ a suitable smoothing perturbation. We also investigate the case in which $D_X$ is not invertible but $0$ is isolated in the $L^2$-spectrum of $D_X$. We prove index formulas relating these delocalized eta invariants to Atiyah-Patodi-Singer delocalized indices on $G$-proper manifolds with boundary. In order to achieve this program we give a detailed account of both the large and small time behaviour of the heat-kernel of perturbed Dirac operators, as a map from the positive real line to the algebra of Lafforgue integral operators. We apply these results to the definition of rho-numbers associated to $G$-homotopy equivalences between closed $G$-proper manifolds and to the study of their bordism properties. We also define delocalized signatures of manifolds with boundary satisfying an invertibility assumption on the differential form Laplacian of the boundary in middle degree and prove an Atiyah-Patodi-Singer formula for these delocalized signatures.