$G$-equivariance of formal models of flag varieties

Abstract

Let $\mathbb{G}$ be a split connected reductive group scheme over the ring of integers $\mathfrak{o}$ of a finite extension $L|{\mathbb{Q}}_p$ and $\lambda \in X(\mathbb{T})$ an algebraic character of a split maximal torus $\mathbb{T}\subseteq \mathbb{G}$. Let us also consider the rigid analytic flag variety $X^{\mathrm{rig}}$ of $\mathbb{G}$ and $G=\mathbb{G}(L)$. In the first part of this paper, we introduce a family of $\lambda$-twisted differential operators on a formal model $\mathfrak{Y}$ of $X^{\mathrm{rig}}$. We compute their global sections and we prove coherence together with several cohomological properties. In the second part, we define the category of coadmissible $G$-equivariant arithmetic $\mathcal{D}(\lambda )$-modules over the family of formal models of the rigid flag variety $X^{\mathrm{rig}}$. We show that if $\lambda$ is such that $\lambda +\rho$ is dominant and regular ($\rho$ being the Weyl character), then the preceding category is anti-equivalent to the category of admissible locally analytic $G$-representations, with central character $\lambda$. In particular, we generalize the main results from a paper by Huyghe, Patel, Schmidt and Strauch (2019) for algebraic characters.