Mathematical Research Letters

Volume 26 (2019)

Number 6

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

Pages: 1677 – 1745



Christine Huyghe (IRMA, Université de Strasbourg, France)

Deepam Patel (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Tobias Schmidt (IRMAR, Université de Rennes 1, Rennes, France)

Matthias Strauch (Department of Mathematics, Indiana University, Bloomington, In., U.S.A.)


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 $\mathbb{X}^{\mathrm{rig}}$.

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