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Grassmanniennes affines tordues sur les entiers

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.