Fundamental lemma of endoscopy

Let $F$ be a nonarchimedean local field, let $G$ be an unramified reductive group, and let $H$ be an unramified endoscopic group. Choose hyperspecial subgroups $K\subset G(F)$ and $K_H\subset H(F)$ , with volume $1$ . The fundamental lemma states that the unit functions

\mathbf 1_K \quad\text{and}\quad \mathbf 1_{K_H}

are matching functions under endoscopic transfer. Equivalently, their matching strongly regular semisimple orbital integrals satisfy the transfer-factor identity.

Hecke-algebra form

More generally, the transfer homomorphism between unramified spherical Hecke algebras, defined through the Satake isomorphism, sends a spherical Hecke function on $G$ to a function on $H$ with matching stable orbital integrals. The unit-element statement is the historically central case.

Why “lemma” is misleading

The statement is a deep theorem. Langlands and Shelstad formulated it as the local identity needed to stabilize the trace formula. Ngô Bảo Châu proved the Lie algebra fundamental lemma by interpreting the relevant orbital integrals through the geometry of the Hitchin fibration. Reduction and transfer results of Waldspurger, Hales, Cluckers–Loeser, and others connect the Lie algebra, group, and characteristic-zero formulations.

Geometric content

Over a function field, orbital integrals become weighted point counts on fibers related to affine Springer fibers. Globally, the Hitchin fibration organizes these fibers. A support theorem and comparison of endoscopic summands in its $\ell$ -adic cohomology yield the required identities.

Variants

The standard fundamental lemma must be distinguished from:

the weighted fundamental lemma needed for the full trace formula;
twisted fundamental lemmas;
the nonstandard fundamental lemma for groups with related root systems;
relative fundamental lemmas for relative trace formulas.

Each has its own matching data and hypotheses.

References

Ngô Bảo Châu, “Le lemme fondamental pour les algèbres de Lie,” Publications Mathématiques de l'IHÉS 111 (2010), 1–169. arXiv.
Ngô Bảo Châu, “Survey on the fundamental lemma.” PDF.