No abstract

Abstract.We prove an analogue of the Beilinson-Bernstein Theorem, in the case of the projective space, for the ring of arithmetical differential operators introduced by P. Berthelot.

Soient p un nombre premier, V un anneau de valuation discrète complet d’inégales caractéristiques (0,p) , et G un groupe réductif et deployé sur \operatorname{Spec}V . Nous obtenons un théorème de localisation, en utilisant les distributions arithmétiques, pour le faisceau des opérateurs différentiels arithmétiques sur la variété de drapeaux formelle de G. Nous donnons une application à la cohomologie rigide pour des ouverts dans la variété de drapeaux en caractéristique p. Let p be a prime number, V a complete discrete valuation ring of unequal characteristics (0,p) , and G a connected split reductive algebraic group over \operatorname{Spec}V . We obtain a localization theorem, involving arithmetic distributions, for the sheaf of arithmetic differential operators on the formal flag variety of G. We give an application to the rigid cohomology of open subsets in the characteristic p flag variety.

Let o be a complete discrete valuation ring of mixed characteristic p0, pq and X 0 a smooth formal o-scheme. Let X Ñ X 0 be an admissible blow-up. In the first part, we introduce sheaves of differential operators D :X,k on X, for every sufficiently large positive integer k, generalizing Berthelot's arithmetic differential operators on the smooth formal scheme X 0 . The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf D X,8 " lim Ð Ýk D :X,k and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves D X,8 , over all admissible blow-ups X, is a sheaf D xX0y on the Zariski-Riemann space of X 0 , which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, i.e., for D : X,k , D X,8 , and D xX0y .

Let G be a connected split reductive group over a finite extension L of Q p , denote by X the flag variety of G, and let G " GpLq. In this paper we prove that formal models X of the rigid analytic flag variety X rig are D :X,k -affine for certain sheaves of arithmetic differential operators D :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 pM X,k q of modules M X,k over D :X,k on the projective system of all formal models X of X rig .CHRISTINE HUYGHE, DEEPAM PATEL, TOBIAS SCHMIDT, AND MATTHIAS STRAUCH 6.3. Principal series representations 56 References 571 These sheaves were denoted r D : n,k in [32] to distinguish them from the sheaves of arithmetic differential operators introduced by P. Berthelot. For ease of notation, we have decided to drop the tilde throughout this paper.