## The fundamental lemma implies the transfer. (Le lemme fondamental implique le transfert.)(French)Zbl 0871.22005

Let $$G$$ be a $$p$$-adic reductive group and $$H$$ an endoscopic group to $$G$$. The conjecture of transfer asserts that there should be a map $$f\mapsto f_H$$ from smooth, compactly supported functions on $$G$$ to those on $$H$$ such that the invariant distributions derived from weighted orbital integrals by J. Arthur [J. Am. Math. Soc. 1, 323-383 (1988; Zbl 0682.10021)] when applied to $$f$$ can be computed by the analogous distributions applied to $$f_H$$. Thus it allows to transfer computations of invariant distributions to endoscopic groups. In [J. Reine Angew. Math. 465, 41-99 (1995; Zbl 0829.11030)] the present author gave, among other things, the analogous conjecture for Lie algebras and conjectured an equality of some weight factors attached to Lie $$G$$ and Lie $$H$$. This weight factor conjecture, if true for all endoscopic groups, implies the transfer conjecture for Lie algebras.
In the present paper the author formulates conditions which guarantee the weight factor conjecture to hold true. As a main result he shows that to a given datum $$(G,H)$$ there is a number field $$k$$ and a global datum $$(\underline G, \underline H)$$ over $$k$$ such that, if the localizations $$(\underline G_v, \underline H_v)$$ satisfy the fundamental lemma for almost all places $$v$$ of $$k$$, then the weight factor conjecture for $$(G,H)$$ is true. The fundamental lemma in this context is the conjectural equality of stable and relative orbital integrals evaluated at the characteristic function of a hyperspecial Lie subalgebra. The proof uses ideas of Langlands and Kottwitz around the stabilization of the trace formula.

### MSC:

 22E35 Analysis on $$p$$-adic Lie groups 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F72 Spectral theory; trace formulas (e.g., that of Selberg)

### Citations:

Zbl 0682.10021; Zbl 0829.11030
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