Orbital integral - Knowlpedia

Let $F$ be a local field, let $G$ be a connected reductive $F$ -group, and let $\gamma\in G(F)$ be semisimple. For a test function $f\in C_c^\infty(G(F))$ , the orbital integral of $f$ at $\gamma$ is

O_\gamma(f) = \int_{G_\gamma(F)\backslash G(F)} f(x^{-1}\gamma x)\,d\dot x,

where $G_\gamma$ is the centralizer of $\gamma$ and the quotient measure comes from chosen Haar measures.

Invariance

The distribution $O_\gamma$ depends only on the $G(F)$ -conjugacy class of $\gamma$ . It is invariant under conjugating the test function. Its numerical value depends on the Haar-measure normalization, so measure choices are part of any exact transfer identity.

For $F$ nonarchimedean, $C_c^\infty$ means locally constant and compactly supported. For an archimedean field it means smooth and compactly supported.

Strongly regular case

When $\gamma$ is strongly regular semisimple, $G_\gamma$ is a torus and the integral has its cleanest form. Different rational conjugacy classes can nevertheless lie in one stable conjugacy class. Their sum is a stable orbital integral, while character-weighted sums are $\kappa$ -orbital integrals.

Global role

For a factorizable adelic test function $f=\bigotimes_v f_v$ , a regular elliptic global orbital term factors, after compatible measure choices, as a centralizer volume times a product of local orbital integrals. The geometric side of the Arthur–Selberg trace formula also contains weighted orbital integrals attached to Levi subgroups and nonelliptic classes.

Lie algebra version

For $X\in\mathfrak g(F)$ semisimple and $f\in C_c^\infty(\mathfrak g(F))$ , one uses the same formula with $x^{-1}Xx=\operatorname{Ad}(x^{-1})X$ . The Lie algebra version is central to geometric proofs of the fundamental lemma.

References

Harish-Chandra, “A submersion principle and its applications,” in Geometry and Analysis, 1981.
Ngô Bảo Châu, “Survey on the fundamental lemma,” §§1.3–1.4. PDF.