Duke Mathematical Journal

Sur l'homologie des fibres de Springer affines tronquées

Pierre-Henri Chaudouard and Gérard Laumon

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Résumé

En suivant les travaux de Goresky, Kottwitz et MacPherson [12], on calcule l'homologie des fibres de Springer affines tronquées dans le cas non ramifié mais sous une hypothèse de pureté. Dans le cas équivalué, on démontre cette hypothèse de pureté. La troncature dépend d'un paramètre qui est un diviseur sur une variété torique -adique. Dans les cas non ramifiés et équivalués, pour chaque fibre de Springer affine tronquée, on définit un faisceau cohérent gradué sur cette variété torique dont l'espace des sections globales s'identifie à l'homologie -adique de la fibre considérée. Pour certaines familles de groupes endoscopiques, ces faisceaux apparaissent dans des suites exactes. On en déduit alors le lemme fondamental pondéré d'Arthur [5, conjecture 5.1] dans les cas équivalués et non ramifiés.

Abstract

Following Goresky, Kottwitz, and MacPherson [12], 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 -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 -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 [5, Conjecture 5.1] in the unramified equivalued case

Article information

Source
Duke Math. J., Volume 145, Number 3 (2008), 443-535.

Dates
First available in Project Euclid: 15 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1229349903

Digital Object Identifier
doi:10.1215/00127094-2008-057

Mathematical Reviews number (MathSciNet)
MR2462112

Zentralblatt MATH identifier
1206.11065

Subjects
Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E67: Loop groups and related constructions, group-theoretic treatment [See also 58D05]
Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 14F63

Citation

Chaudouard, Pierre-Henri; Laumon, Gérard. Sur l'homologie des fibres de Springer affines tronquées. Duke Math. J. 145 (2008), no. 3, 443--535. doi:10.1215/00127094-2008-057. https://projecteuclid.org/euclid.dmj/1229349903


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