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Register a new userDecomposition au-dessus des param`etres de Langlands elliptiques
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We prove that, over any elliptic global Langlands parameter $sigma$, the cuspidal cohomology groups of moduli stacks of shtukas are given by a formula involving a finite dimensional representation of the centralizer of $sigma$. It is a first step in the direction of Arthur-Kottwitz conjectures.
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We study the reduction modulo $l$ of some elliptic representations; for each of these representations, we give a particular lattice naturally obtained by parabolic induction in giving the graph of extensions between its irreducible sub-quotient of its reduction modulo $l$. The principal motivation for this work, is that these lattices appear in the cohomology of Lubin-Tate towers.
This is the memoir of my habilitation thesis, defended on March 29 th, 2013 (Universite Paris XI).
The principal result of this work is the freeness in the $ overline{mathbb Z}_l$-cohomology of the Lubin-Tate tower. The strategy is of global nature and relies on studying the filtration of stratification of the perverse sheaf of vanishing cycles of some Shimura varieties of Kottwitz-Harris-Taylor types, whose graduates can be explicited as some intermediate extension of some local system constructed in the book of Harris andTaylor. The crucial point relies on the study of the difference between such extension for the two classical $t$-structures $p$ and $p+$. The main ingredients use the theory of derivative for representations of the mirabolic group.
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We establish a positive characteristic analogue of intersection cohomology for polarized variations of Hodge structure. This includes: a) the decomposition theorem for the intersection de Rham complex; b) the $E_1$-degeneration theorem for the intersection de Rham complex of a periodic de Rham bundle: c) the Kodaira vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle.
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