Worked solution: Ngô's proof via the Hitchin fibration and geometric support theorem (2008)

Fix a nonarchimedean local field $F$ with ring of integers $\mathcal{O}$, a reductive group $G$ unramified over $F$, and its hyperspecial maximal compact $K=G(\mathcal{O})$. For a regular semisimple $\gamma\in G(F)$ the orbit $G(F)/C_\gamma(F)\cdot\gamma$ carries an invariant measure, and the orbital integral $O_\gamma(\mathbf{1}_K)=\int_{G(F)/C_\gamma(F)}\mathbf{1}_K(x^{-1}\gamma x)\,dx$ is the volume of the set of $x$ conjugating $\gamma$ into $K$. Because $H^1(F,C_\gamma)$ splits the stable conjugacy class of $\gamma$ into finitely many rational conjugacy classes, one sums over them weighted by a character $\kappa$ to form the $\kappa$-orbital integral $O_\gamma^\kappa(\mathbf{1}_K)$.

The endoscopic group $H$ is a quasi-split group built from $(G,\gamma,\kappa)$ whose dual group is a subgroup of ${}^LG$ containing the centraliser data; its conjugacy classes are designed to correspond to the stable classes of $G$. The fundamental lemma, conjectured by Langlands and Shelstad in 1983, asserts the transfer identity $O_\gamma^\kappa(\mathbf{1}_{K}) = \Delta(\gamma,\gamma^H)\,SO_{\gamma^H}(\mathbf{1}_{K^H})$, where $SO$ on the right is a stable orbital integral — a sum over all rational classes in one stable class — and $\Delta(\gamma,\gamma^H)$ is an explicit transfer factor accounting for the mismatch between the two measures.

The identity is the exact local input needed to compare the geometric sides of the Arthur–Selberg trace formulas of $G$ and $H$: term-by-term matching of orbital integrals is what stabilises the trace formula and powers Langlands functoriality (e.g. transfer of automorphic representations to $H$). Orbital integrals are genuinely hard $p$-adic integrals — before Ngô, only rank-one groups and a few small cases had been checked by hand, so the conjecture stood for roughly 25 years; the first simplification, Waldspurger's reduction to Lie algebras, is the next step.