On the index of Dirac operators on arithmetic quotients

TL;DR

This paper uses the Arthur-Selberg trace formula to compute the index of Dirac operators on arithmetic quotients, showing simplifications in certain cases where orbital integrals vanish, aligning with compact case results.

Contribution

It provides a new expression for the Dirac operator index on arithmetic quotients using trace formulas, with specific cases where orbital integrals vanish, simplifying the index calculation.

Findings

Orbital integrals vanish for products of rank one spaces in the Euler operator case.

Index formula reduces to the compact case when orbital integrals are zero.

Trace formula approach effectively computes Dirac operator indices on arithmetic quotients.

Abstract

Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient over a totally real field with at least two real embeddings as the integral over the index form plus a sum of orbital integrals. For the Euler operator these orbital integrals are shown to vanish for products of rank one spaces. In this case the index theorem looks exactly as in the compact case.