Lefschetz formula for locally symmetric spaces

Abstract

Let $G$ be a semi-simple real Lie group of real rank one and $Γ$ be a discrete subgroup in $G$ such that $Γ\backslash G$ has finite volume. We introduce a new group $C^*$-algebra $C^*_r(G, Γ)$, which provides a natural framework for defining index classes of Dirac-type operators on the locally symmetirc space $Γ\backslash G /K$. We show that Dirac operators define elements in the $K$-theory of $C^*_r(G, Γ)$ and use Hecke correspondences to study their Lefschetz numbers. Our main result is an explicit formula for the Lefschetz number of Hecke operators.