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

TL;DR

This paper introduces delocalized eta invariants for signature operators on G-proper manifolds, relating them to index formulas, rho-numbers, and signatures, advancing the understanding of spectral invariants in equivariant geometry.

Contribution

It defines new delocalized eta invariants for perturbed Dirac operators on G-proper manifolds and establishes their relation to index theory and bordism invariants.

Findings

Defined delocalized eta invariants for perturbed Dirac operators.

Proved index formulas relating these invariants to delocalized indices.

Applied results to rho-numbers and delocalized signatures in equivariant topology.

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.