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The fundamental lemma for Lie algebras. (Le lemme fondamental pour les algèbres de Lie.) (French) Zbl 1200.22011

It has been proved by Waldspurger that the fundamental lemma and transfer of orbital integrals for reductive groups over a non-archimedean local field can be deduced from the fundamental lemma for Lie algebras. Waldspurger also proved that validity of this fundamental lemma in the case of equal characteristics implies the validity in the case of unequal characteristics. The present article gives a proof in the case of equal characteristic \(p\) greater than twice the Coxeter number of the group.
The fundamental lemma has its origin in the problem of stabilisation of the trace formula for a reductive group over a global field, of characteristic \(p\) in the present case. It is an equality between a certain linear combination of local orbital integrals (a \(\kappa\)-orbital integral) for the adjoint representation of \(G\) and an orbital integral (a stable orbital integral) for an endoscopic group \(H\) of \(G\).
The method is geometric and uses local and global arguments. Locally, one has the affine Springer fibers, introduced by Kazhdan and Lusztig. Stable orbital integrals and \(\kappa\)-orbital integrals can be expressed in terms of numbers of points in quotients of affine Springer fibers. Via Grothendieck’s Lefschetz trace formula one obtains a cohomological interpretation of these integrals. As a global analogue of the affine Springer fibers, one has the fibers of the Hitchin fibration. An important tool here is the action on the space of the Hitchin fibration of a Picard stack over the base of the Hitchin fibration. A product formula gives the relation between the Hitchin fiber and the affine Springer fibers.
Geometric stabilisation is now formulated in terms of the perverse cohomology of the Hitchin fibration, or more precisely, of its restriction on a certain étale open subset (it is not the exact translation of the stabilisation of the trace formula).
The author proves a result on the support of the simple constituents of the perverse sheaves. The geometric stabilisation is first performed on a still smaller open subset, where the support theorem can be applied. Using this global result, the fundamental lemma is proved. Using the fundamental lemma, the geometric stabilisation is proved completely.
The author also proves a conjecture of Waldspurger (“the nonstandard lemma”), which, together with the fundamental lemma for Lie algebras, implies the twisted fundamental lemma.
The article is very well written.

MSC:

22E57 Geometric Langlands program: representation-theoretic aspects
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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