A Beilinson–Bernstein Theorem for Twisted Arithmetic Differential Operators on the Formal Flag Variety

Andrés Sarrazola-Alzate

doi:10.5802/jtnb.1272

Andrés Sarrazola-Alzate ¹

¹ Calle 23 AA Sur Nro. 5-200, Kilómetro 2+200 Variante al Aeropuerto José María Córdova, Envigado-Antioquia. Universidad EIA, oficina B105. Código Postal: 055428 Colombia

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 1-43.

Résumé
Abstract

Soit $ℚ_{p}$ le corps des nombres $p$ -adiques et $𝔾$ un schéma en groupes réductif, connexe et déployé sur $ℤ_{p}$ . Nous introduirons un faisceau d’opérateurs différentiels arithmétiques tordus sur la variété des drapeaux formelle de $𝔾$ , associée à un caractère général. En particulier, nous généraliserons les résultats de [21], concernant la $𝒟^{†}$ -affinité de la variété des drapeaux formelle lisse de $𝔾$ , de certains gerbes d’opérateurs différentiels arithmétiques tordus $p$ -adiquement complets, associés à un caractère algébrique, et les résultats de [24] concernant le calcul des sections globales.

Let $ℚ_{p}$ be the field of $p$ -adic numbers and $𝔾$ a split connected reductive group scheme over $ℤ_{p}$ . In this work we will introduce a sheaf of twisted arithmetic differential operators on the formal flag variety of $𝔾$ , associated to a general character. In particular, we will generalize the results of [21], concerning the $𝒟^{†}$ -affinity of the smooth formal flag variety of $𝔾$ , of certain sheaves of $p$ -adically complete twisted arithmetic differential operators associated to an algebraic character, and the results of [24] concerning the calculation of the global sections.

Reçu le : 2019-10-07
Révisé le : 2023-06-17
Accepté le : 2023-07-12
Publié le : 2024-07-25

DOI : 10.5802/jtnb.1272

Classification : 22E50, 14L30, 13N10, 32C38
Mots clés : Formal flag variety, twisted arithmetic differential operators, Beilinson–Bernstein correspondence

Affiliations des auteurs :

Andrés Sarrazola-Alzate ¹

¹ Calle 23 AA Sur Nro. 5-200, Kilómetro 2+200 Variante al Aeropuerto José María Córdova, Envigado-Antioquia. Universidad EIA, oficina B105. Código Postal: 055428 Colombia

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A {Beilinson{\textendash}Bernstein} {Theorem} for {Twisted} {Arithmetic} {Differential} {Operators} on the {Formal} {Flag} {Variety}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Andrés Sarrazola-Alzate. A Beilinson–Bernstein Theorem for Twisted Arithmetic Differential Operators on the Formal Flag Variety. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 1-43. doi : 10.5802/jtnb.1272. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1272/

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