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On the homology of truncated affine Springer fibers. (Sur l’homologie des fibres de Springer affines tronquées.) (French) Zbl 1206.11065

Summary: Following M. Goresky, R. Kottwitz and R. MacPherson [Duke Math. J. 121, No. 3, 509–561 (2004; Zbl 1162.14311)], we compute the homology of truncated affine Springer fibers in the unramified case but under a purity assumption. We prove this assumption in the equivalued case. The truncation parameter is viewed as a divisor on an \(\ell\)-adic toric variety. In the unramified and equivalued cases, for each truncated affine Springer fiber, we introduce a graded coherent sheaf on the toric variety whose space of global sections is precisely the \(\ell\)-adic homology of this fiber. Moreover, for some families of endoscopic groups, these sheaves show up in an exact sequence. As a consequence, we prove Arthur’s weighted fundamental lemma [J. Inst. Math. Jussieu 1, No. 2, 175–277 (2002; Zbl 1040.11038), Conjecture 5.1] in the unramified equivalued case

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
14M15 Grassmannians, Schubert varieties, flag manifolds
22E67 Loop groups and related constructions, group-theoretic treatment
11R39 Langlands-Weil conjectures, nonabelian class field theory
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