## Endoscopy and change of the characteristic. (Endoscopie et changement de caractÃ©ristique.)(French)Zbl 1102.22010

Let $$G$$ be a connected reductive group over a non-archimedean local field $$F$$ and let $$H$$ be an endoscopic group for $$G$$. Assume $$G$$ and $$H$$ are unramified (i.e. quasi-split over $$F$$ and split over an unramified extension of $$F$$). The fundamental lemma for $$G$$ and $$H$$ in its Lie algebra version is a conjectural equality between certain linear combinations of orbital integrals on $${\mathfrak g}(F)$$ and stable orbital integrals on $${\mathfrak h}(F)$$, where $${\mathfrak g}$$ and $${\mathfrak h}$$ are the Lie algebras of $$G$$ and $$H$$.
In the present article this fundamental lemma is transformed into an equality in which $$F$$ appears only through its residue field $${\mathbb F}_q$$ (under the assumption that the characteristic $$p$$ of $${\mathbb F}_q$$ is big enough). Thus, if $$F$$ and $$F'$$ have the same residue field and the fundamental lemma is true for $$F$$, then it is true for $$F'$$. This is particularly interesting when $$\text{char}(F)=p$$ and $$\text{char}(F')=0$$.
Because the proof is by induction (replace $$G$$ by the centralizer of a semisimple element), a more general situation must be considered: $$G$$ and $$H$$ are assumed to be quasi-split over $$F$$ and split over the maximal tamely (= moderately) ramified extension $$F^{\text{mod}}$$ of $$F$$. Together with $$G$$, the author considers the twists of $$G$$ by cocycles from $$H^{1}(\text{Gal}(F^{\text{mod}}/F),G(F^{\text{mod}}))$$.
Let the residue field $${\mathbb F}_q$$ be fixed. The Galois group $$\text{Gal}(F^{\text{mod}}/F)$$ can be identified with a group $$\Gamma$$ which is independent of $$F$$. Fix a root datum $${\mathcal D}$$ provided with an action of $$\Gamma$$. Given $$F$$, we can associate to $${\mathcal D}$$ a moderately ramified group $$G$$ over $$F$$. There is also a finite group $$D$$ attached to $${\mathcal D}$$ and for all $$F$$ a bijection between $$D$$ and $$H^{1}(\text{Gal}(F^{\text{mod}}/F),G(F^{\text{mod}}))$$. We then have the twisted groups $$G_d,d\in D$$. The main theorem of the article gives an expression for the orbital integrals on $${\mathfrak g}_d$$ $$(d\in D)$$ in terms independent of $$F$$.
Among the constructions which have to be made independent of $$F$$ there are the following. First, the building of $$G_d$$ over $$F$$. It is shown that this building can be described in such a way that the quotients of the Moy-Prasad filtration become independent of $$F$$. This makes the test functions independent of $$F$$. The conjugacy classes present a different problem. The author develops an abstract construction in which the set of ordinary conjugacy classes in a stable conjugacy class is in bijection with a set independent of $$F$$. Here appears the necessity to consider the groups $$G_d$$ together.
Moreover the proof of the main theorem demands an extensive study of the centralizers of semisimple elements.
For the stable orbital integrals on $${\mathfrak h}_d$$ $$(d\in {\mathcal D}_H)$$ there is an analogous expression, independent of $$F$$.
Specializing the result to the case of unramified $${\mathcal D}$$ and $${\mathcal D}_H$$ we get the assertion concerning the fundamental lemma stated above.

### MSC:

 22E35 Analysis on $$p$$-adic Lie groups 20G25 Linear algebraic groups over local fields and their integers
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