Weighted orbital integral

Definition

Let $G$ be a connected reductive group over a local field, let $M$ be a Levi subgroup, and let $\gamma\in M(F)$ be regular enough for the integral below. An Arthur weighted orbital integral has the form

J_M(\gamma,f)= \int_{G_\gamma(F)\backslash G(F)} f(x^{-1}\gamma x)\,v_M(x)\,dx.

Here $v_M(x)$ is a nonnegative weight built from the convex hull of the Iwasawa height vectors of $x$ for the parabolic subgroups with Levi $M$ . When $M=G$ , the weight is $1$ and $J_G(\gamma,f)$ is the ordinary orbital integral.

Why the weight appears

Arthur truncation cuts a noncompact automorphic kernel in several parabolic directions. After unfolding, the volume of the resulting truncation polytope produces $v_M(x)$ . Weighted orbital integrals therefore encode contributions induced from proper Levi subgroups on the fine geometric side of the trace formula.

Invariance and transfer

The initial distributions $J_M(\gamma,\cdot)$ are generally noninvariant. Arthur combines them with correction terms to form invariant distributions $I_M(\gamma,\cdot)$ . Stabilizing these terms requires weighted endoscopic transfer, which is subtler than the unweighted fundamental lemma.

References

James Arthur, “An introduction to the trace formula,” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings 4, 2005, §§11 and 18–19. Clay.
James Arthur, “The local behaviour of weighted orbital integrals,” Duke Mathematical Journal 56 (1988), 223–293.