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Twisted affine Grassmannians over the integers Pereira Lourenço, João Nuno


Let $G$ be a quasi-split reductive connected group over $\mathbb{Q}(t)$ which splits over $\mathbb{Q}(\zeta_e, t^{1/e})$, $e=2$ or $3$, whose derived group is absolutely simple simply connected and whose maximal torus corresponds to a sum of permutation modules of rank 1 or $e$. Fixing a maximal split torus $S$ of $G$ and a facet $\mathbf{f}$ of the apartment corresponding to $S$ in the building of $G$ over $\mathbb{Q}((t))$, we construct a smooth, affine and connected group scheme over $\mathbb{Z}[t]$, which should be regarded as a family of parahoric group schemes of type $\mathbf{f}$ in varying characteristics, generalising previous work of Pappas-Zhu over $\mathbb{Z}[1/e][t]$ and Tits over $\mathbb{Z}[t,t^{-1}]$. Since in the critical characteristic $e$ the group becomes generally pseudo-reductive, we briefly explain how Bruhat-Tits theory of reductive groups over local fields extends to the pseudo-reductive setting. Finally, we consider the local and global affine Grassmannians of the $\mathbb{Z}[t]$ group scheme and prove their representability by an ind-projective ind-scheme as well as normality of Schubert varieties. Time permitting, we discuss the resulting local models obtained in wildly ramified cases and their relation with the diamond local models of Scholze.

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