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In this article we define and study a Harder-Narasimhan filtration of finite flat group schemes over an unequal characteristic valuation ring. We also study that type of filtrations for families of finite flat group schemes parametrised by formal schemes. [ABSTRACT FROM AUTHOR]

Abstract: We study the Hurwitz action of the classical braid group on factorisations of a Coxeter element c in a well-generated complex reflection group W. It is well known that the Hurwitz action is transitive on the set of reduced decompositions of c in reflections. Our main result is a similar property for the primitive factorisations of c, i.e. factorisations with only one factor which is not a reflection. The motivation is the search for a geometric proof of Chapoton''s formula for the number of chains of given length in the non-crossing partitions lattice . Our proof uses the properties of the Lyashko–Looijenga covering and the geometry of the discriminant of W. [Copyright &y& Elsevier]

Soient p un nombre premier, F une extension finie de Qp, et ψ un caractère additif non trivial de F. La correspondance de Langlands donne une bijection σ ↦ π (σ) entre les représentations Φ–semisimples de dimension n du groupe de Weil–Deligne de F, à isomorphisme près, et les représentations lisses irréductibles de , à isomorphisme près. Pour certaines représentations r du groupe dual de , on sait associer, par voie globale, à une représentation lisse irréductible π de , des facteurs L(π, r, s) et ɛ (π, r, s, ψ). On conjecture l’égalité L(π (σ), r, s) = L (r ○ σ, s), et de même pour les facteurs ɛ, quand σ est une représentation de dimension n du groupe de Weil–Deligne de F. Répondant à une question de D. Jiang et D. Soudry, nous prouvons que si r = Λ2 ou , on a L(π (σ), r, s) = L (r ○ σ, s) et ɛ (π (σ), r, s, ψ) = α ɛ (r ○ σ, s, ψ), où α est une racine de l’unité. [ABSTRACT FROM PUBLISHER]

Berkovich (Invent. Math. 125(2):367–390, ), Fujiwara (Duke Math. J. 80(1):15–57, ) and Huber (J. Algebraic Geom. 7(2):359–403, ) have proved that the fiber of the vanishing cycles at a point of the special fiber depends only on the formal completion at this point. We refine this result and prove the invariance under formal completion of the perverse monodromy filtration on the fiber of vanishing cycles. This result is used in an essential way by Boyer (Invent. Math. doi:, ). [ABSTRACT FROM AUTHOR]

In the geometric situation of the simple Shimura varieties of Kottwitz studied in Harris and Taylor’s book (The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, ), we describe the monodromy filtration of the vanishing cycles complex and the spectral sequence associated to it. We prove in particular that this filtration coincides with the weight one up to shift. Thanks to the Berkovich-Fargues’ theorem, we deduce the description of the local monodromy filtration of the Deligne-Carayol model. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Springer fiber corresponding to a nilpotent endomorphism u of nilpotent order 2. We establish a necessary and sufficient condition of singularity for the components of . To cite this article: L. Fresse, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: Let X be a complex algebraic variety. We define and study new theories of characteristic classes, defined on the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. One of them, is a homology class version of the motivic measure and generalizes the corresponding Hirzebruch characteristic. It unifies the Chern class transformation of Schwartz and MacPherson, the Todd class transformation of Baum–Fulton–MacPherson and the L-class transformation of Cappell–Shaneson. To cite this article: J.-P. Brasselet et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]