Endoscopic transfer factor

Let $H$ be the endoscopic group in an endoscopic datum for $G$ . For related strongly regular semisimple elements $\gamma_H\in H(F)$ and $\delta\in G(F)$ , an endoscopic transfer factor is a nonzero complex scalar

\Delta(\gamma_H,\delta)

used to compare stable orbital integrals on $H$ with weighted sums of orbital integrals on $G$ .

With one standard convention, matching test functions satisfy

SO_{\gamma_H}(f^H) = \sum_{\delta} \Delta(\gamma_H,\delta)\,O_\delta(f),

where the sum runs over rational conjugacy classes in the matching stable class. Equivalently the right side is a normalized $\kappa$ -orbital integral.

What the factor corrects

A $\kappa$ -orbital integral depends on the chosen representative of a stable class, whereas the left side is stable. The transfer factor changes by the inverse scalar needed to cancel that dependence. It also records root-theoretic discriminants and Galois-cohomological pairings.

Construction data

The Langlands–Shelstad factor is assembled from terms customarily denoted

\Delta_I\Delta_{II}\Delta_{III}\Delta_{IV}.

Its construction uses choices such as splittings, $a$ -data, $\chi$ -data, and compatible Haar measures. Absolute normalizations can instead be fixed by a Whittaker datum or by rigid inner-twist data. The resulting transfer identity is independent of auxiliary choices when all terms are normalized coherently.

Convention warning

Sources differ by inverses, complex conjugation, arithmetic versus geometric normalization, and the use of $\Delta'$ or $\Delta_D$ in twisted endoscopy. A formula containing a transfer factor is not meaningful until the normalization of orbital integrals and local Langlands correspondence is also stated.

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, “On splitting invariants and sign conventions in endoscopic transfer,” 2012. arXiv.