## 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:

 2.2e+36 Analysis on $$p$$-adic Lie groups 2.2e+51 Representations of Lie and linear algebraic groups over local fields

### Keywords:

fundamental lemma; weighted orbital integrals

Zbl 1184.22003
Full Text:

### References:

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