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Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties. (Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples.) (French) Zbl 1172.14016
Let $$K$$ be a finite extension of $${\mathbb Q}_p$$ with ring of integers $${\mathcal O}_K$$. For some fixed positive integer $$d$$, we consider the group $$D^\times_{K,d}$$ of invertibles of the central division algebra over $$K$$ with invariant $$\frac{1}{d}$$ an the Weil group $$W_K$$ of $$K$$.
Using étale cohomology, Deligne has constructed a series of representations $${\mathcal U}^i_{K,l,d}$$ of the group $$GDW_K(d):=GL_d(K)\times D^\times_{K,d}\times W_K$$. Now let $$\rho$$ be any irreducible representation of $$D^\times_{K,d}$$ such that the corresponding $$\pi:=JL(\rho)$$ is a cuspidal representation of $$GL_d(K)$$. The first result of the paper under review calculates the $$\rho$$-isotypical components $${\mathcal U}^i_{K,l,d}(\rho)$$ of $${\mathcal U}^i_{K,l,d}$$ completely via globalizing the investigations of [M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Annals of Mathematics Studies 151. (Princeton), NJ: Princeton University Press. (2001; Zbl 1036.11027)]: There, the authors study Shimura varieties of PEL type defined over some CM field F, which are associated to a particular group $$G/{\mathbb Q}$$ of similitudes. To certain subgroups $$I\subset G({\mathbb A}^\infty)$$ one associates a tower of Shimura varieties $$X_{\mathcal I}:=(X_I)_{I\in{\mathcal I}}\to Spec({\mathcal O}_v)$$, where $$v$$ is some prime above $$p$$ such that the completion $$F_v$$ of the localization of $$F$$ in $$v$$ is isomorphic to $$K$$, on which $$G({\mathbb A}^\infty)$$ acts via correspondences.
The special geometric fiber $$\overline{X}_{\mathcal I}$$ of $$X_{\mathcal I}$$ is stratified by locally closed Hecke schemes $$\overline{X}_{\mathcal I}^{=h}$$ for $$1\leq d\leq h$$. In the present paper, methods of Harris and Taylor to describe the restriction of the sheaf of vanishing cycles to these strata are generalized.
Now let $$\Psi_{\mathcal I}:=R\Psi_{\eta_v}({\overline{\mathbb Q}}_l)[d-1](\frac{d-1}{2})$$ be the complex of vanishing cycles on $$\overline{X}_{\mathcal I}$$ viewed as a complex of $$W_v$$-perverse Hecke sheaves, and for an irreducible cuspidal representation $$\pi_v$$ of $$GL_g(F_v)$$ denote $$\Psi_{{\mathcal I},\pi_v}$$ the corresponding isotypical component. The second main result of the present article describes its bi-graded pieces with respect to kernels and images of the pro-nilpotent monodromy operator $$N$$ in the category of $$W_{F_v}$$-perverse sheaves on $$\overline{X}_{\mathcal I}$$.
The proof works by induction on the $${\mathcal U}^i_{F_v,l,h}$$ for $$1\leq h < d$$ (supposed to be known) of the Deligne-Carayol model. The initial step is comprised by former work of the author [P. Boyer, Invent. Math. 138, No. 3, 573–629 (1999; Zbl 1161.11408)]. The comparison theorem of Berkovich and Fargues is then applied in order to obtain the result except for those perverse sheaves with support in supersingular points. This case finally is treated using methods from Harris and Talyor’s work again in connection with the auto-duality of the local model with respect to the Zelevinski involution.
The article is nicely written, and also includes an appendix on Hecke schemes and their associated sheaves for the readers’ convenience.

##### MSC:
 14G22 Rigid analytic geometry 14G35 Modular and Shimura varieties 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11G35 Varieties over global fields 11R39 Langlands-Weil conjectures, nonabelian class field theory 14L05 Formal groups, $$p$$-divisible groups 11G45 Geometric class field theory 11Fxx Discontinuous groups and automorphic forms
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