Fundamental lemma of the Langlands programme

For a reductive group $G$ over a $p$-adic (or function) field and an endoscopic group $H$ attached to it, the fundamental lemma asserts that the orbital integral of the characteristic function of a hyperspecial maximal compact subgroup of $G$ equals, up to an explicit transfer factor $\Delta(\gamma,\gamma_H)$, the corresponding $\kappa$-orbital integral on $H$: schematically $O_\gamma(\mathbf{1}_K) = \Delta(\gamma,\gamma_H)\,O_{\gamma_H}^{\kappa}(\mathbf{1}_{K_H})$.

Ngô Bảo Châu proved the fundamental lemma for Lie algebras of arbitrary reductive groups by relating orbital integrals to the cohomology of the Hitchin fibration and establishing a geometric support theorem; standard reduction arguments (Waldspurger; Cluckers–Hales–Loeser) then extend the result to the general and group cases.