Lefschetz formula for locally symmetric spaces

TL;DR

This paper develops a new $C^*$-algebra framework for analyzing Dirac operators on locally symmetric spaces of rank one, deriving an explicit Lefschetz formula for Hecke operators using $K$-theory and index theory.

Contribution

It introduces a novel $C^*$-algebra approach to define and study index classes of Dirac operators on these spaces, leading to a Lefschetz formula for Hecke operators.

Findings

Defined a new $C^*$-algebra $C^*_r(G, \Gamma)$ for locally symmetric spaces.

Established that Dirac operators define $K$-theory elements in this algebra.

Derived an explicit Lefschetz formula for Hecke operators.

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 $C^*$-algebra $C^*_r(G, \Gamma)$, 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 $C^*_r(G, \Gamma)$ and use Hecke correspondences to study their Lefschetz numbers. Our main result is an explicit formula for the Lefschetz number of Hecke operators.