Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo

The geometric Satake equivalence in mixed characteristic

par Prof. Peter SCHOLZE (Université de Bonn & IHES)

Europe/Paris
Centre de conférences Marilyn et James Simons (IHES)

Centre de conférences Marilyn et James Simons

IHES

Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
Description

In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa(_p). These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec(). The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.

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