##
**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.

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.

Reviewer: J. G. M. Mars (Utrecht)

### MSC:

22E35 | Analysis on \(p\)-adic Lie groups |

20G25 | Linear algebraic groups over local fields and their integers |