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On the fundamental lemma for standard endoscopy: reduction to unit elements. (English) Zbl 0840.22032
The fundamental lemma for standard endoscopy follows from the matching of unit elements in Hecke algebras. A simple form of the stable trace formula, based on the matching of unit elements, shows the fundamental lemma to be equivalent to a collection of character identities. These character identities are established by comparing them to a compact-character expansion of orbital integrals. L. Clozel has deduced the fundamental lemma for stable base change from the corresponding result for the unit elements of Hecke algebras. This paper adapts Clozel’s argument to standard endoscopy, thereby reducing this fundamental lemma for reductive groups to the unit elements of Hecke algebras. Unlike stable base change, the matching of units is not currently known.
One of the main purposes of this paper is to clarify the set of local conditions that imply the fundamental lemma. These local conditions are formalized as local data. Local arguments reduce the fundamental lemma to groups $$G$$ with connected anisotropic centers. For such $$G$$, local data are a collection of finite character identities between a reductive group $$G$$ and an endoscopic group $$H$$. If local data exist, and if the fundamental lemma is known for the Levi factors of $$G$$, then the fundamental lemma holds for $$G$$. Although the conditions are local, the only methods currently known to establish the existence of local data are global. The final section shows how a simple stable trace formula may be used to prove the existence of local data. This argument assumes the matching of unit elements of Hecke algebras at almost all places of a global situation constructed in this paper.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 22E35 Analysis on $$p$$-adic Lie groups 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 20G25 Linear algebraic groups over local fields and their integers
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