Endoscopic datum - Knowlpedia

Let $G$ be a connected reductive group over a local or global field $F$ . An endoscopic datum for $G$ is, in a commonly used shorthand, a triple

\mathfrak e=(H,s,\eta),

where $H$ is a quasi-split connected reductive $F$ -group, $s\in\widehat G$ is semisimple, and

\eta:{}^L H\longrightarrow {}^L G

$\eta$ is an $L$ -homomorphism between $L$ -groups, identifying the dual group $\widehat H$ with the identity component $\operatorname{Cent}(s,\widehat G)^\circ$ of a centralizer, subject to the standard Galois-action and central conditions. Data are taken up to an equivalence relation involving $\widehat G$ -conjugacy.

Full form of the datum

In general the relevant extension need not be presented as the ordinary $L$ -group of $H$ . One therefore writes

(H,\mathcal H,s,\xi),

where $\mathcal H$ is an extension of the Weil group by $\widehat H$ and $\xi:\mathcal H\to{}^LG$ realizes the prescribed centralizer. The shorter triple is appropriate only after an $L$ -embedding or auxiliary $z$ -pair has been chosen.

Endoscopic group

The group $H$ is usually not a subgroup of $G$ . Its dual group is a connected centralizer inside $\widehat G$ , so the relation is naturally visible on the dual side. Matching strongly regular stable conjugacy classes of $H(F)$ map to stable classes of $G(F)$ .

The datum is elliptic when the connected component of the relevant Galois-fixed center of $\widehat H$ , modulo that of $\widehat G$ , is trivial. Elliptic data govern discrete terms.

Purpose

Endoscopic data index correction terms in the stable trace formula. A transfer factor and matching functions compare stable orbital integrals on $H$ with $\kappa$ -orbital integrals on $G$ . On the spectral side this becomes a relation among packet characters.

Ordinary versus twisted endoscopy

Twisted endoscopy adds an automorphism or a twisted space and is essential for transfers from classical groups to general linear groups. Its datum and transfer factors contain extra structure; an ordinary datum should not be silently used in the twisted setting.

References

Robert P. Langlands and Diana Shelstad, “On the definition of transfer factors,” Mathematische Annalen 278 (1987), 219–271. DOI.
Robert E. Kottwitz and Diana Shelstad, Foundations of Twisted Endoscopy, Astérisque 255, 1999. Numdam.