Reduction to Lie algebra version of fundamental lemma?

For the purpose of this answer let us say that "fundamental lemma" means "fundamental lemma for the unit of the unramified Hecke algebra". I do not think that "FL for Lie algebras => FL for groups" was proved in "Le lemme fondamental implique le transfert". This implication is however proved in greater generality (twisted endoscopy) in the paper "L'endoscopie tordue n'est pas si tordue", also by Waldspurger (see Conjecture 2 there; I think the non-standard FL for Lie algebras was also proved by Ngo). Just below Conjecture 2 Waldspurger writes that Hales "A simple definition of transfer factors for unramified groups" gives the implication in the case of standard endoscopy, but I do not think that this is completely obvious. In Hales' paper there is a reduction to the case of topologically unipotent elements, and I think that for these elements you can deduce the FL from the case of Lie algebras via the exponential map. There are detailed proofs in Waldspurger's paper for the twisted case; presumably they can be slightly simplified for the ordinary case. Note that the actual reduction in section 5.12 is not very long, so you can probably get a more precise idea of the proof than "use the exponential" without reading the whole paper.

Of course Ngo's theorem is about equicharacteristic local fields, whereas the version of FL for Lie algebras needed here is for characteristic zero local fields (and indeed the exponential map is used!). There are two ways to "switch fields": a paper of Waldspurger, or using model theory (I imagine you can find all the relevant references in Ngo's paper).

You can find it in Waldspurger's paper titled Le lemme fondamental implique le transfert.

DOI: https://doi.org/10.1023/A:1000103112268