The affine Grassmannian as a presheaf quotient

Kęstutis Česnavičius

doi:10.5802/crmath.736

Research article - Algebraic geometry

The affine Grassmannian as a presheaf quotient

Kęstutis Česnavičius ¹

¹ CNRS, Université Paris-Saclay, Laboratoire de mathématiques d’Orsay, F-91405, Orsay, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 523-532.

Abstract (Original language)
Résumé

For a reductive group $G$ over a ring $A$, its affine Grassmannian $\operatorname{Gr}_G$ plays important roles in a wide range of subjects and is typically defined as the étale sheafification of the presheaf quotient $LG/L^+G$ of the loop group $LG$ by its positive loop subgroup $L^+G$. We show that the Zariski sheafification gives the same result. Moreover, for totally isotropic $G$ (for instance, for quasi-split $G$), we show that no sheafification is needed at all: $\operatorname{Gr}_G$ is already the presheaf quotient $LG/L^+G$, which seems new already in the classical case of $\operatorname{GL}_n$ over $\mathbb{C}$. For totally isotropic $G$, we also show that the affine Grassmannian may be formed using polynomial loops. We deduce all of these results from the study of $G$-torsors on $\mathbb{P}^1_A$ that is ultimately built on the geometry of $\operatorname{Bun}_G$.

Pour un groupe réductif $G$ sur un anneau $A$, sa grassmannienne affine $\mathrm{Gr}_G$ joue un rôle important dans un grand nombre de sujets et est typiquement définie comme la faisceautisation étale du quotient de préfaisceaux $LG/L^+G$ du groupe de lacets $LG$ par son sous-groupe d’arcs $L^+G$. Nous montrons que la faisceautisation de Zariski donne le même résultat. De plus, pour $G$ totalement isotrope (par exemple, pour $G$ quasi-déployé), nous montrons qu’aucune faisceautisation n’est nécessaire : $\mathrm{Gr}_G$ est déjà le quotient de préfaisceaux $LG/L^+G$, ce qui semble nouveau déjà dans le cas classique d’un $G$ sur $\mathbb{C}$. Pour $G$ totalement isotrope, nous montrons aussi que sa grassmannienne affine peut être formée en utilisant des lacets polynomiaux. Nous déduisons tous ces résultats de l’étude des torseurs de $G$ sur $\mathbb{P}^1_A$ basé sur la géométrie de $\mathrm{Bun}_G$.

Received: 2024-01-13
Revised: 2025-01-02
Accepted: 2025-02-19
Published online: 2025-05-26

DOI: 10.5802/crmath.736

Classification: 14L15, 14M17
Keywords: Affine Grassmannian, loop group, reductive group, torsor
Mots-clés : Grassmannienne affine, groupe de lacets, groupe réductif, torseur

Author's affiliations:

Kęstutis Česnavičius ¹

¹ CNRS, Université Paris-Saclay, Laboratoire de mathématiques d’Orsay, F-91405, Orsay, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {The affine {Grassmannian} as a presheaf quotient},
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Kęstutis Česnavičius. The affine Grassmannian as a presheaf quotient. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 523-532. doi : 10.5802/crmath.736. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.736/

References
Cited by

[1] Schémas en groupes (SGA 3). Tome III. Structure des schémas en groupes réductifs (Philippe Gille; Patrick Polo, eds.), Documents Mathématiques, 8, Société Mathématique de France, 2011, lvi+337 pages

[2] Tom Bachmann Affine Grassmannians in $𝔸^{1}$ -homotopy theory, Sel. Math., New Ser., Volume 25 (2019) no. 2, 25, 14 pages | DOI | MR | Zbl

[3] Alexander Beilinson; Vladimir Drinfeld Quantization of Hitchin’s integrable system and Hecke eigensheaves (Online at https://math.uchicago.edu/~drinfeld/langlands/QuantizationHitchin.pdf)

[4] Bhargav Bhatt Algebraization and Tannaka duality, Camb. J. Math., Volume 4 (2016) no. 4, pp. 403-461 | DOI | MR | Zbl

[5] Bhargav Bhatt; Daniel Halpern-Leistner Tannaka duality revisited, Adv. Math., Volume 316 (2017), pp. 576-612 | DOI | MR | Zbl

[6] Alexis Bouthier; Kęstutis Česnavičius Torsors on loop groups and the Hitchin fibration, Ann. Sci. Éc. Norm. Supér., Volume 55 (2022) no. 3, pp. 791-864 | DOI | MR | Zbl

[7] Kęstutis Česnavičius Grothendieck–Serre in the quasi-split unramified case, Forum Math. Pi, Volume 10 (2022), e9, 30 pages | DOI | MR | Zbl

[8] Kęstutis Česnavičius Problems About Torsors over Regular Rings, Acta Math. Vietnam., Volume 47 (2022) no. 1, pp. 39-107 | DOI | MR | Zbl

[9] Kęstutis Česnavičius; Roman Fedorov Unramified Grothendieck–Serre for isotropic groups (2023) | arXiv

[10] Kęstutis Česnavičius; Alex Youcis The analytic topology suffices for the $B_{dR}^{+}$ -Grassmannian (2024) (To appear in Proceedings of the Simons Symposium on p-adic Hodge theory) | arXiv

[11] Antoine Chambert-Loir; Johannes Nicaise; Julien Sebag Motivic integration, Progress in Mathematics, 325, Birkhäuser, 2018, xx+526 pages | DOI | MR

[12] Roman Fedorov On the Grothendieck–Serre conjecture about principal bundles and its generalizations, Algebra Number Theory, Volume 16 (2022) no. 2, pp. 447-465 | DOI | MR | Zbl

[13] Mathieu Florence; Philippe Gille Residues on affine Grassmannians, J. Reine Angew. Math., Volume 776 (2021), pp. 119-150 | DOI | MR | Zbl

[14] Alexander Grothendieck; Jean Dieudonné Éléments de géométrie algébrique. I. Le langage des schémas, Publ. Math., Inst. Hautes Étud. Sci. (1960) no. 4, 228 pages | MR

[15] Alexander Grothendieck; Jean Dieudonné Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math., Inst. Hautes Étud. Sci. (1967) no. 32, 361 pages | Zbl

[16] Ivan Panin; Anastasia Stavrova On the Gille theorem for the relative projective line: I (2023) | arXiv

[17] Ivan Panin; Anastasia Stavrova On the Gille theorem for the relative projective line (2024) | arXiv

[18] Georgios Pappas; Michael Rapoport Twisted loop groups and their affine flag varieties, Adv. Math., Volume 219 (2008) no. 1, pp. 118-198 | DOI | MR | Zbl

[19] Michel Raynaud; Laurent Gruson Critères de platitude et de projectivité. Techniques de « platification » d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | DOI | MR | Zbl

[20] The Stacks Project Authors Stacks Project http://stacks.math.columbia.edu

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