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The fundamental lemma for unitary groups. (Le lemme fondamental pour les groupes unitaires.) (French) Zbl 1179.22019

This article gives a proof of the fundamental lemma for unitary groups of rank \(n\) over a local field of equal characteristics \(>n\). The study is based on the geometric interpretation of elliptic endoscopy given by Ngô, which uses the Hitchin fibration. In the article the Hitchin fibration is described explicitly for \(G=\text{U}(n)\), as well as its relation with the Hitchin fibration for an endoscopic group \(H=\text{U}(n_1)\times \text{U}(n_2)\) for \(G\), where \(n=n_1+n_2\). (For general reductive groups see B. C. Ngô [Invent. Math. 164, No. 2, 399–453 (2006; Zbl 1098.14023)].) From a result on the perverse cohomology of the Hitchin fibration a numerical identity is deduced, using a fixed-point theorem. Explicit computation of the factors in this global identity gives the fundamental lemma.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22E35 Analysis on \(p\)-adic Lie groups
14H60 Vector bundles on curves and their moduli
14F20 Étale and other Grothendieck topologies and (co)homologies

Citations:

Zbl 1098.14023
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