Grassmanniennes affines tordues sur les entiers

TL;DR

This paper generalizes the theory of twisted affine Grassmannians to more complex group structures over integers, establishing foundational properties like normality and coherence in these extended settings.

Contribution

It extends the construction of twisted affine Grassmannians to wildly ramified, quasi-split, and residually split groups with induced tori, introducing new models and proving key geometric properties.

Findings

Proved normality of Schubert varieties in the new setting.

Established Zhu's coherence theorem for these affine Grassmannians.

Constructed smooth, affine, connected parahoric group schemes over $Z[t]$.

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.