$\mathscr{D}^{\dagger}$-affinity of formal models of flag varieties

Let $\mathbb{G}$ be a connected split reductive group over a finite extension $L$ of $\mathbb{Q}_p$, denote by $\mathbb{X}$ the flag variety of $\mathbb{G}$, and let $G = \mathbb{G}(L)$. In this paper we prove that formal models $\mathfrak{X}$ of the rigid analytic flag variety $\mathbb{X}^{\mathrm{rig}}$ are $\mathscr{D}^{\dagger}_{\mathfrak{X}, k}$‑affine for certain sheaves of arithmetic differential operators $\mathscr{D}^{\dagger}_{\mathfrak{X}, k}$. Furthermore, we show that the category of admissible locally analytic $G$-representations with trivial central character is naturally anti-equivalent to a full subcategory of the category of $G$-equivariant families $(\mathscr{M}_{\mathfrak{X}, k})$ of modules $(\mathscr{M}_{\mathfrak{X}, k})$ over $\mathscr{D}^{\dagger}_{\mathfrak{X}, k}$ on the projective system of all formal models $\mathfrak{X}$ of $\mathbb{X}^{\mathrm{rig}}$.

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D.P. would like to acknowledge support from IHÉS and the ANR program $p$‑adic Hodge Theory and beyond (ThéHopaD) ANR-11-BS01-005. T.S. would like to acknowledge support of the Heisenberg programme of Deutsche Forschungsgemeinschaft (SCHM 3062/1-1). M.S. would like to acknowledge the support of the National Science Foundation (award DMS-1202303).

Received 3 May 2018

Accepted 12 June 2018

Published 6 March 2020