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

Waldspurger proved in the 1990s that the fundamental lemma for a reductive group $G$ follows formally from the corresponding statement for its Lie algebra $\mathfrak{g} = \mathrm{Lie}(G)$: an orbital integral on $G$ near the identity can be matched, term by term, with an orbital integral on $\mathfrak{g}$ (Hales 2011, §1; Waldspurger 1997). This is a genuine simplification because $\mathfrak{g}$ is a vector space, so its geometry (centralizers, discriminants) is governed by ordinary polynomial algebra rather than group multiplication.

Waldspurger also showed, by comparing 'close' local fields, that it suffices to prove the Lie-algebra statement over local fields of positive equal characteristic such as $\mathbb{F}_q((t))$, rather than over the $p$-adic numbers $\mathbb{Q}_p$ directly; a decade later Raf Cluckers, Thomas Hales, and François Loeser gave an independent, more flexible proof of essentially the same transfer using the model theory of motivic integration, which also handles the $\kappa$-weighted and group versions needed at the end of the proof.

These two reductions are exactly what let Ngô Bảo Châu attack the fundamental lemma using algebraic geometry over a curve defined over a finite field: everything from here on takes place for Lie algebras in equal positive characteristic.