Geometric stabilisation via p-adic integration

TL;DR

This paper presents a new proof of Ngô's Geometric Stabilisation Theorem using p-adic integration, linking Hitchin fibre cohomology for reductive groups and endoscopy groups without relying on the Decomposition and Support Theorem.

Contribution

The authors provide a novel proof of the Geometric Stabilisation Theorem based on p-adic integration techniques, extending duality results to quasi-split groups and describing the inertia stack in terms of endoscopic data.

Findings

New proof of Ngô's Geometric Stabilisation Theorem

Description of inertia stack via endoscopic data

Extension of duality for Hitchin fibres to quasi-split groups

Abstract

In this article we give a new proof of Ng\^o's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_{\kappa}$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.