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The weighted fundamental lemma. II: Cohomological statements. (Le lemme fondamental pondéré. II: Énoncés cohomologiques.) (French. English summary) Zbl 1264.11043

In this article the authors continue the presentation of their proof of Arthur’s weighted fundamental lemma. The subject is, first of all, the study of the perverse cohomology of the truncated Hitchin fibration introduced in the preceding article [Compos. Math. 146, 1416–1506 (2010; Zbl 1206.14026)]. One important result is the “support theorem”: on a certain open subset \({\mathcal A}^{bon}\) of the base \({\mathcal A}\) of the fibration the perverse cohomology sheaves are direct sums of perverse extensions of local systems on open subsets of \({\mathcal A}^{bon}\cap {\mathcal A}^{ell}\), where \({\mathcal A}^{ell}\) is the elliptic part of \(\mathcal A\). The main cohomological result relates the perverse cohomology of the truncated Hitchin fibration for \(G\) to that for endoscopic groups of \(G\). By the support theorem the proof is reduced to Ngô’s theorem on the elliptic part of the cohomology, see B. C. Ngô’s article [Publ. Math. IHES 111, 1–169 (2010; Zbl 1200.22011)], where the ordinary fundamental lemma is proved.
By computations of the trace of the Frobenius action on the perverse cohomology sheaves using the Grothendieck-Lefschetz trace formula, a global form of the weighted fundamental lemma is proved. From this global result the local lemma is deduced.
In the present article the reductive groups are assumed to be split. The authors promise to treat the general case later.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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