On smoothing singularities of elliptic orbital integrals on GL(n) and Beyond Endoscopy

TL;DR

This paper extends Altu's method of smoothing elliptic orbital integrals from GL(2) to GL(n) and discusses potential applications to other reductive groups within Langlands' Beyond Endoscopy framework.

Contribution

It generalizes the smoothing technique for elliptic orbital integrals to higher rank groups and explores its applicability to arbitrary reductive groups.

Findings

Extension of smoothing method to GL(n)

Discussion of obstructions for Poisson summation in general groups

Potential implications for Beyond Endoscopy

Abstract

Recent work of Altu\u{g} continues the preliminary analysis of Langlands' Beyond Endoscopy proposal for $GL(2)$ by removing the contribution of the trivial representation to the trace formula using a Poisson summation formula. We show that Altu\u{g}'s method of smoothing real elliptic orbital integrals by an approximate functional equation extends to $GL(n)$. We also discuss the case of an arbitrary reductive group, and remaining obstructions for applying Poisson summation.