On elliptic factors in real endoscopic transfer I

TL;DR

This paper investigates the structure and transfer of packets of representations for real groups, introducing new tools and reinterpretations to improve endoscopic transfer compatibility with orbital integral transfer, especially for limits of discrete series.

Contribution

It introduces novel splittings based on fundamental maximal tori and explicit realizations of Levi groups, enhancing the understanding of endoscopic transfer and spectral factors for real groups.

Findings

Reinterpreted Adams-Johnson transfer via spectral Langlands-Shelstad factors

Provided explicit realizations of Levi groups from dual data

Simplified definitions of twisted endoscopic transfer in critical cases

Abstract

This paper is concerned with the structure of packets of representations and some refinements that are helpful in endoscopic transfer for real groups. It includes results on the structure and transfer of packets of limits of discrete series representations. It also reinterprets the Adams-Johnson transfer of certain nontempered representations via spectral analogues of the Langlands-Shelstad factors, thereby providing structure and transfer compatible with the associated transfer of orbital integrals. The results come from two simple tools introduced here. The first concerns a family of splittings of the algebraic group G under consideration; such a splitting is based on a fundamental maximal torus of G rather than a maximally split maximal torus. The second concerns a family of Levi groups attached to the dual data of a Langlands or an Arthur parameter for the group G. The introduced splittings provide explicit realizations of these Levi groups. The tools also apply to maps on stable conjugacy classes associated with the transfer of orbital integrals. In particular, they allow for a simpler version of the definitions of Kottwitz-Shelstad for twisted endoscopic transfer in certain critical cases. The paper prepares for spectral factors in twisted endoscopic transfer that are compatible in a certain sense with the standard factors discussed here. This compatibility is needed for Arthur's global theory. The twisted factors themselves will be defined in a separate paper.