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Endoscopy and change of characteristic: weighted orbital integrals. (Endoscopie et changement de caractéristique: intégrales orbitales pondérées.) (French) Zbl 1184.22003

Let \(F\) be a non-Archimedean locally compact field, \(G\) a connected reductive \(F\)-group and \(H\) an endoscopic group of \(G\). The Langlands functoriality principle predicts a correspondence between the automorphic representations of \(G\) and those of \(H\). Over the years, the stable trace formula approach to establish the correspondence was developed to the extent that the only piece missing is the weighted fundamental lemma conjecture – identities between weighted orbital integrals on \(G\) and weighted orbital integrals on \(H\).
This paper establishes the following important result: if the weighted fundamental lemma conjecture holds over a field \(F_1\) with residue field \(F_q\), then it holds for a field \(F_2\) with the same residue field (when \(q\) is large enough). Ngo Bao Chau recently established the fundamental lemma conjecture over a function field. It is likely that Ngo’s method can be extended to give a proof of the weighted fundamental lemma conjecture over a function field. Then the result of this paper gives the weighted fundamental lemma conjecture over a field of characteristic zero as well.
The proof is a delicate extension of the method introduced in the author’s previous work [ibid. 59, 1753–1818 (2009; Zbl 1184.22003)], where the corresponding result for the fundamental lemma was established. The author notes that there is a very different approach, by Cluckers and Loeser, that establishes the same results using logic.

MSC:

22E35 Analysis on \(p\)-adic Lie groups
22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 1184.22003
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References:

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