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Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration.

For the purposes of the trace formula, one actually needs the fundamental lemma for the group. In his ICM notes, Ngo states that Waldspurger shows how to reduce the group statement to the Lie algebra statement, but the reference is not clear (to me at least).

My question is

Where is this reduction discussed? A precise reference would be great.

In lieu of the reference, I would also gladly accept a sketch of the reduction:

How does this reduction work?

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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).

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  • $\begingroup$ This is fantastic. Thank you for the helpful answer! $\endgroup$ – WSL Sep 28 '19 at 8:40
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    $\begingroup$ Both transferring between all t.n. elements in the Lie algebra and all t.u. elements in the group, and transferring between equi-characteristic and mixed-characteristic fields, require that $p$ be very large (the bound in the latter case being totally ineffective). Is that just the state of things now, that this reduction is only available for very large $p$, or are there ways around it? $\endgroup$ – LSpice Sep 30 '19 at 1:24
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You can find it in Waldspurger's paper titled Le lemme fondamental implique le transfert.

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

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  • $\begingroup$ I will happily accept once I can verify that the result is in that paper, but can't find it on a key word search. This is exactly the paper that (it appears that) Ngo references. Evidently I will need to study the paper, unless you can point me to the result. $\endgroup$ – WSL Sep 5 '19 at 15:20

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