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

Sarrazola-Alzate, Andrés

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

Let ${\mathbb{Q}}_p$ be the field of $p$-adic numbers and $𝔾$ a split connected reductive group scheme over ${\mathbb{Z}}_p$. 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.

Soit ${\mathbb{Q}}_p$ le corps des nombres $p$-adiques et $𝔾$ un schéma en groupes réductif, connexe et déployé sur ${\mathbb{Z}}_p$. 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.

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

[1] Ardakov, Konstantin; Wadsley, Simon On irreducible representations of compact $p$-adic analytic groups, Ann. Math., Volume 178 (2013) no. 2, pp. 453-557 | Zbl | DOI

[2] Beilinson, Alexander; Bernstein, Joseph Localisation de $𝔤$-modules, C. R. Math. Acad. Sci. Paris, Volume 292 (1981), pp. 15-18 | Zbl

[3] Beilinson, Alexander; Bernstein, Joseph A proof of Jantzen conjectures, I. M. Gelfand Seminar. Part 1: Papers of the Gelfand seminar in functional analysis held at Moscow University, Russia, September 1993 (Advances in Soviet Mathematics), Volume 16, American Mathematical Society, 1993, pp. 1-50 | Zbl | MR

[4] Beilinson, Alexander; Ginzburg, Victor Wall-crossing functors and $𝒟$-modules, Represent. Theory, Volume 3 (1999), pp. 1-31 | Zbl | DOI | MR

[5] Berthelot, Pierre $𝒟$-modules arithmétiques I. Opérateurs différentiels de niveau fini, Ann. Sci. Éc. Norm. Supér., Volume 29 (1996) no. 2, pp. 185-272 | DOI | Zbl | MR

[6] Berthelot, Pierre Introduction à la theorie arithmetique des $𝒟$-modules, Cohomologies $p$-adiques et applications arithmetiques. II (Astérisque), Volume 279, SMF, 2002, pp. 1-80 | Zbl

[7] Berthelot, Pierre; Ogus, Arthur Notes on crystalline cohomology. (MN-21), Mathematical Notes, 21, Princeton University Press, 2015 | DOI | MR

[8] Bezrukavnikov, Roman; Mirković, Ivan; Rumynin, Dmitriy Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. Math., Volume 167 (2008) no. 3, pp. 945-991 | DOI | Zbl

[9] Björk, Jan-Erik Analytic $𝒟$-modules and applications, Mathematics and its Applications (Dordrecht), 247, Springer, 2013, xiii+581 pages | Zbl | MR

[10] Borho, Walter; Brylinski, Jean-Luc Differential operators on homogeneous spaces. I: Irreducibility of the associated variety for annihilators of induced modules, Invent. Math., Volume 69 (1982), pp. 437-476 | DOI | Zbl | MR

[11] Borho, Walter; Brylinski, Jean-Luc Differential operators on homogeneous spaces. II: relative enveloping algebras, Bull. Soc. Math. Fr., Volume 117 (1989) no. 2, pp. 167-210 | DOI | Numdam | Zbl | MR

[12] Brylinski, Jean-Luc; Kashiwara, Masaki Démonstration de la conjecture de Kazhdan-Lusztig sur les modules de Verma, C. R. Math. Acad. Sci. Paris, Volume 291 (1980), pp. 373-376 | Zbl

[13] Demazure, Michel; Gabriel, Pierre Groupes algébriques. Tome I: Géométrie algébrique. Généralités. Groupes commutatifs, Masson; North-Holland, 1970 | Zbl | MR

[14] Dixmier, Jacques Enveloping algebras, North-Holland Mathematical Library, 14, North-Holland, 1977 | DOI | MR

[15] Grothendieck, Alexander Éléments de géométrie algébrique. I: Le langage des schémas, Publ. Math., Inst. Hautes Étud. Sci., Volume 4 (1960), pp. 1-228 | Numdam | Zbl | MR

[16] Grothendieck, Alexander Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie). Rédigé avec la colloboration de J. Dieudonné, Publ. Math., Inst. Hautes Étud. Sci., Volume 32 (1967), pp. 1-361 | Zbl

[17] Grothendieck, Alexander; Dieudonné, Jean Éléments de géométrie algébrique. III: Étude cohomologique des faisceaux cohérents (seconde partie), Publ. Math., Inst. Hautes Étud. Sci., Volume 17 (1963), pp. 137-223 | Zbl

[18] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 2013 | MR

[19] Hotta, Ryoshi; Takeuchi, Kiyoshi; Tanisaki, Toshiyuki $𝒟$-modules, perverse sheaves and representation theory, Progress in Mathematics, 236, Springer, 2007 | MR

[20] Huyghe, Christine ${𝒟}^{†}$-affinité de l’espace projectif, Compos. Math., Volume 108 (1997) no. 3, pp. 277-318 | DOI | Zbl | MR

[21] Huyghe, Christine Un théorème de Beilinson-Bernstein pour les $𝒟$-modules arithmétiques, Bull. Soc. Math. Fr., Volume 137 (2009) no. 2, pp. 159-183 | DOI | Zbl | MR

[22] Huyghe, Christine; Patel, Deepam; Schmidt, Tobias; Strauch, Matthias ${𝒟}^{†}$-affinity of formal models of flag varieties, Math. Res. Lett., Volume 26 (2019) no. 6, pp. 1677-1745 | DOI | Zbl | MR

[23] Huyghe, Christine; Schmidt, Tobias $𝒟$-modules arithmétiques, distributions et localisation, Rend. Semin. Mat. Univ. Padova, Volume 139 (2018), pp. 1-76 | DOI | Zbl | MR

[24] Huyghe, Christine; Schmidt, Tobias $𝒟$-modules arithmétiques sur la variété de drapeaux, J. Reine Angew. Math., Volume 754 (2019), pp. 1-15 | DOI | Zbl | MR

[25] Huyghe, Christine; Schmidt, Tobias; Strauch, Matthias Arithmetic structures for differential operators on formal schemes, Nagoya Math. J., Volume 243 (2021), pp. 157-204 | DOI | Zbl | MR

[26] Jantzen, Jen C. Representation of algebraic groups, American Mathematical Society, 2007 | DOI | MR

[27] Kazhdan, David; Lusztig, George Representations of Coxeter groups and Hecke algebras, Invent. Math., Volume 53 (1979), pp. 165-184 | DOI | Zbl | MR

[28] Li, Huishi; Van Oystaeyen, Freddy Zariskian filtrations, 2, Kluwer Academic Publishers, 1996, ix+252 pages | DOI | MR

[29] Lipman, Joseph; Hashimoto, Mitsuyasu Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, 1960, Springer, 2009 | DOI

[30] Miličić, Dragan Localization and representation theory of reductive Lie groups (1993) (http://www.math.utah.edu/milicic/Eprints/book.pdf)

[31] Milne, James S. Etale cohomology, Princeton Mathematical Series, 33, Princeton University Press, 1980, xiii+323 pages | DOI | MR

[32] Mumford, David; Fogarty, John; Kirwan, Frances Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, 34, Springer, 1994 | DOI | MR

[33] Patel, Deepam; Schmidt, Tobias; Strauch, Matthias Locally analytic representations and sheaves on the Bruhat-Tits building, Algebra Number Theory, Volume 8 (2014) no. 6, pp. 1365-1445 | DOI | Zbl | MR

[34] Patel, Deepam; Schmidt, Tobias; Strauch, Matthias Locally analytic representations of $\mathrm{GL}(2,L)$ via semistable models of ${\mathbb{P}}^1$, J. Inst. Math. Jussieu, Volume 18 (2019) no. 1, pp. 125-187 | DOI | Zbl | MR

[35] Sarrazola-Alzate, Andrés $G$-equivariance of formal models of flag varieties, Rend. Semin. Mat. Univ. Padova, Volume 150 (2019), pp. 1-79 | DOI | Zbl | MR

[36] Schneider, P.; Teitelbaum, J. Algebras of $p$-adic distribution and admissible representations, Invent. Math., Volume 153 (2003) no. 1, pp. 145-196 | DOI | Zbl | MR

[37] Van Der Bergh, Michael Some generalities on $G$-equivariant quasi-coherent ${𝒪}_X$ and ${𝒟}_X$-modules (1994) (Preprint Université Louis Pasteur, Strasbourg)