Knowlpedia

Stable orbital integral

The sum of orbital integrals over rational conjugacy classes in one stable conjugacy class.

Let FF be a and let γ∈G(F)\gamma\in G(F) be . The stable orbital integral of f∈Cc∞(G(F))f\in C_c^\infty(G(F)) is

SOγ(f)=∑γ′Oγ′(f),SO_\gamma(f) = \sum_{\gamma'} O_{\gamma'}(f),

where γ′\gamma' ranges over representatives of the G(F)G(F)-conjugacy classes in the of γ\gamma, and the use compatibly transported on their centralizer tori.

Stability

Unlike an individual orbital integral, SOγSO_\gamma depends only on the stable conjugacy class. A assembled from such expressions vanishes on whose stable orbital integrals all vanish.

The displayed unweighted sum is the basic strongly regular definition. Extensions to singular elements and global trace formulas can involve Kottwitz signs, measures, or limiting procedures; those conventions must be stated.

Kappa refinements

The rational classes inside the stable class are parametrized by a finite AγA_\gamma. For a κ:Aγ→C×\kappa:A_\gamma\to\mathbb C^\times, the associated weights the summands by κ\kappa. The stable orbital integral is the case of the trivial character.

Endoscopic role

For an HH, transfer seeks a function fHf^H such that stable orbital integrals on HH equal transfer-factor-weighted κ\kappa-orbital integrals on GG. The proves this matching for the unramified unit elements, and general transfer provides matching test functions.

References
  1. Robert P. Langlands and Diana Shelstad, “On the definition of transfer factors,” Mathematische Annalen 278 (1987), 219–271. DOI.
  2. Ngô Bảo Châu, “Survey on the fundamental lemma,” §2.2. PDF.