Lefschetz formula for locally symmetric spaces

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