Kappa-orbital integral

Let $\gamma\in G(F)$ be strongly regular semisimple over a local field. Write

A_\gamma= \ker\!\left[H^1(F,G_\gamma)\to H^1(F,G)\right],

which is a kernel in nonabelian Galois cohomology and parametrizes the rational conjugacy classes in the stable class of $\gamma$ . Given a character $\kappa:A_\gamma\to\mathbb C^\times$ , the $\kappa$ -orbital integral is

O_\gamma^\kappa(f) = \sum_{\gamma'} \kappa(\operatorname{inv}(\gamma,\gamma'))\, O_{\gamma'}(f),

with compatible centralizer measures.

Dependence on a basepoint

The invariant $\operatorname{inv}(\gamma,\gamma')$ uses $\gamma$ as a basepoint. Replacing that basepoint multiplies the distribution by a scalar. The transfer factor compensates for this dependence in endoscopic transfer.

For the trivial character $\kappa=1$ , the expression is the stable orbital integral.

Dual interpretation

Tate–Nakayama duality relates characters of the cohomological group $A_\gamma$ to component-group data in the dual centralizer. The corresponding semisimple dual-group element determines an endoscopic group. Thus the Fourier decomposition of ordinary orbital terms across a stable class produces the endoscopic pieces of the trace formula.

Convention warning

Some sources sum over a larger $H^1(F,G_\gamma)$ , insert Kottwitz signs, or write the character inversely. The choice of $\operatorname{inv}(\gamma,\gamma')$ , transfer-factor normalization, and Haar measures must be read as one package.

References

Robert E. Kottwitz, “Stable trace formula: cuspidal tempered terms,” Duke Mathematical Journal 51 (1984), 611–650. DOI.
Ngô Bảo Châu, “Survey on the fundamental lemma,” §2.2. PDF.