Higher Lefschetz formulas on Γ-proper manifolds

Abstract

Let $Γ$ be a finitely generated discrete group acting properly and cocompactly on a smooth manifold M. By employing heat-kernel techniques we prove a geometric formula for the pairing of the index class associated to a $Γ$-equivariant Dirac operator $D$ with a delocalized cyclic cocycles $τ$ in $HP^\bullet (\mathbb{C}Γ,\langle γ\rangle)$. Our formula takes place on the fixed point manifold $M^γ$ and should be regarded as a higher Lefschetz formula for $D$. The formula involves the Atiyah-Segal-Singer form and an explicit $Z_γ$-invariant form on $M^γ$ that is naturally associated to $τ\in HP^\bullet (\mathbb{C}Γ,\langle γ\rangle)$