Grassmanniennes affines tordues sur les entiers

Abstract

We extend the work of Pappas-Rapoport-Zhu on twisted affine Grassmannians to wildly ramified, quasi-split, and residually split groups, assuming the maximal torus is induced. This relies on the construction, inspired by Tits, of certain smooth, affine, and connected $\mathbb{Z}[t]$-groups of parahoric type, which should be regarded as $\mathbb{Z}$-families of parahoric group schemes, and naturally extends a similar construction in the above articles after inverting $e$. The resulting ${\mathbb{F}}_p(t)$-groups are pseudo-reductive possibly non-standard in the sense of Conrad--Gabber--Prasad, and their ${\mathbb{F}}_p[[t]]$-models are parahoric in our generalized sense. We study their affine Grassmannians, establishing normality of Schubert varieties and Zhu's coherence theorem.

82 pages, in French. The paper was almost completely rewritten and now treats non reduced root systems as well. Applications to the Scholze-Weinstein conjecture on local models were left out for future work