Transfert d'int\'egrales orbitales pour le groupe m\'etaplectique

TL;DR

This paper develops an endoscopic formalism for the metaplectic group, establishing transfer factors and the fundamental lemma, thus extending prior work to a broader class of groups using advanced descent methods.

Contribution

It introduces a formalism of endoscopy for the metaplectic group, including stable conjugacy, transfer factors, and the fundamental lemma, generalizing previous results for real groups.

Findings

Transfer of antigenuine orbital integrals established

Fundamental lemma for spherical Hecke algebra units proven under certain conditions

Extension of endoscopic transfer theory to metaplectic groups

Abstract

This paper develops a formalism of endoscopy for the metaplectic group. We define the notions of stable conjugacy, elliptic endoscopic groups, correspondence of semisimple geometric conjugacy classes and the transfer factors in this setting, then we establish the transfer of antigenuine orbital integrals. Under the hypothesis that the residual characteristic of F is sufficiently large, the fundamental lemma for the units of the spherical Hecke algebra also holds. This generalizes the prior works of J. Adams and D. Renard for real metaplectic groups. Our approach is based on Harish-Chandra's descent method and the non-standard endoscopy on Lie algebras.