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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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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 at 15:20

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