Lefschetz formula for locally symmetric spaces

Abstract

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

arXiv commentThe previous discussion on KK-theory and C*algebras has been removed, but the main results remain unchanged