\(G\)-equivariance of formal models of flag varieties (Q6082477)

Summary: 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 \(\mathcal{Y}\) 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.