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THÉORIE DE BRUHAT-TITS POUR LES GROUPES QUASI-RÉDUCTIFS

Part of: Algebraic groups Linear algebraic groups and related topics Special varieties Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  25 January 2021

João Lourenço Open the ORCID record for João Lourenço [Opens in a new window]
João Lourenço*
Affiliation:
Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115Bonn, Deutschland ([email protected])
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Abstract

Soient K un corps discrètement valué et hensélien, ${\mathcal {O}}$ son anneau d’entiers supposé excellent, $\kappa $ son corps résiduel supposé parfait et G un K-groupe quasi-réductif, c’est-à-dire lisse, affine, connexe et à radical unipotent déployé trivial. On construit l’immeuble de Bruhat-Tits ${\mathcal {I}}(G, K)$ pour $G(K)$ de façon canonique, améliorant les constructions moins canoniques de M. Solleveld sur les corps locaux, et l’on associe un ${\mathcal {O}}$ -modèle en groupes ${\mathcal {G}}_{\Omega }$ de G à chaque partie non vide et bornée $\Omega $ contenue dans un appartement de ${\mathcal {I}}(G,K)$ . On montre que les groupes parahoriques ${\mathcal {G}}_{\textbf {f}}$ attachés aux facettes peuvent être caractérisés en fonction de la géométrie de leurs grassmanniennes affines, ainsi que dans la thèse de T. Richarz. Ces résultats sont appliqués ailleurs à l’étude des grassmanniennes affines tordues entières.

Keywords

grassmanniennes affinesschémas en groupesthéorie de Bruhat-Titsgroupes parahoriquesgroupes pseudo-réductifsgroupes quasi-réductifsmodèles de Néron

MSC classification

Primary: 14L15: Group schemes
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds 20G25: Linear algebraic groups over local fields and their integers 20E42: Groups with a $BN$-pair; buildings
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 4 , July 2022 , pp. 1331 - 1362
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Achet, R.. Picard group of unipotent groups, restricted Picard functor. 2019. hal-02063586v2.Google Scholar
Achet, R.. Unirational algebraic groups. 2019. hal-02358528.Google Scholar
Anantharaman, S.. Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1. Mémoires de la S.M.F., tome 33 (1973), p. 5–79.Google Scholar
Beilinson, A. & Drinfeld, V.. Quantization of Hitchin's integrable system and Hecke eigensheaves. http://math.uchicago.edu/{~}drinfeld/langlands/hitchin/BD-hitchin.pdf.Google Scholar
Bhatt, B. & Scholze, P.. Projectivity of the Witt vector affine Grassmannian . Invent. Math. 209 (2017), no. 2, 329–423.CrossRefGoogle Scholar
Borel, A. & Tits, J.. Théorèmes de structure et de conjugaison pour les groupes algébriques linéaires . C. R. Acad. Sci., tome 287 (1978), 55–57.Google Scholar
Bosch, S., Lütkebohmert, W. & Raynaud, M.. Néron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21, Berlin, New York: Springer Verlag (1990).CrossRefGoogle Scholar
Bruhat, F. & Tits, J.. Groupes réductifs sur un corps local : I. Données radicielles valuées. Publ. math. de l’I.H.É.S., tome 41 (1972), p. 5-251.Google Scholar
Bruhat, F. & Tits, J.. Groupes réductifs sur un corps local : II. Schémas en groupes. Existence d’une donnée radicielle valuée . Publ. math. de l’I.H.É.S., tome 60 (1984), p.~5–184.CrossRefGoogle Scholar
Conrad, B.. Finiteness theorems for algebraic groups over function fields . Compos. Math 148 (2012), 555–639.CrossRefGoogle Scholar
Conrad, B., Gabber, O. & Prasad, G.. Pseudo-reductive groups. New Mathematical Monographs 26. Cambridge: Cambridge University Press (ISBN 978-1-107-08723-1/hbk; 978-1-316-09243-9/ebook). xxiv, 665 p. (2015).CrossRefGoogle Scholar
Conrad, B. & Prasad, G.. Classification of pseudo-reductive groups. Annals of Mathematics Studies 191, Princeton University Press (2016).Google Scholar
Haines, T., Lourenço, J. & Richarz, T.. On the normality of Schubert varieties: remaining cases in positive characteristic. En préparation. https://arxiv.org/abs/1806.11001. Google Scholar
Haines, T. & Rapoport, M.. On parahoric subgroups . Adv. Math. 219 (2008), 188–198.CrossRefGoogle Scholar
Landvogt, E.. A compactification of the Bruhat-Tits building. Lect. Notes in Math., vol. 1619, Springer-Verlag, Berlin, 1996.Google Scholar
Lang, S.. On quasi algebraic closure . Annals of Mathematics, Second Series, Vol. 55, No. 2 (1952), 373–390.CrossRefGoogle Scholar
Lourenço, J.. Grassmanniennes affines tordues sur les entiers. https://arxiv.org/abs/1912.11918.Google Scholar
Lourenço, J.. Théorie de Bruhat-Tits, grassmanniennes affines et modèles locaux. Dissertation. Rheinische Friedrich-Wilhelms-Universität Bonn (2020).Google Scholar
Mayeux, A., Richarz, T. & Romagny, M.. Torsors under Néron blow-ups. https://arxiv.org/abs/2001.03597.Google Scholar
Oesterlé, J.. Nombres de Tamagawa et groupes unipotents en caractéristique p . Inventiones mathematicae 78 (1984): p. 13–88.CrossRefGoogle Scholar
Pappas, G. & Rapoport, M.. Twisted loop groups and their affine flag varieties . Adv. Math. 219, No. 1, 118–198 (2008).CrossRefGoogle Scholar
Pappas, G. & Zhu, X.. Local models of Shimura varieties and a conjecture of Kottwitz . Invent. Math. 194, No. 1, 147–254 (2013).CrossRefGoogle Scholar
Prasad, G.. A new approach to unramified descent in Bruhat-Tits theory. https://arxiv.org/abs/1611.07430.Google Scholar
Raynaud, M.. Faisceaux amples sur les schémas en groupes et les espaces homogènes. Lecture Notes in Mathematics, Vol. 119, Springer-Verlag, Berlin, 1970.CrossRefGoogle Scholar
Richarz, T.. Affine Grassmannians and geometric Satake equivalences . Int. Math. Res. Not., Vol. 2016, Issue 12, 3717–3767 (2016).CrossRefGoogle Scholar
Richarz, T.. Erratum to “Affine Grassmannians and geometric Satake equivalences”. Int. Math. Res. Not., rnz210 (2019).CrossRefGoogle Scholar
Rosengarten, Z.. Tate duality in positive dimensions over function fields. https://arxiv.org/pdf/1805.00522.pdf.Google Scholar
Schneider, P. & Stuhler, U.. Representation theory and sheaves on the Bruhat-Tits building . Publ. math. de l’IHÉS, tome 85 (1997), 97–191.CrossRefGoogle Scholar
SGA 3. Schémas en groupes. III : Structure des schémas en groupes réductifs. Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 153, Springer-Verlag, Berlin, 1962/1964.Google Scholar
Solleveld, M.. Pseudo-reductive and quasi-reductive groups over non-archimedean local fields . Journal of Algebra 510 (2018), 331–392.CrossRefGoogle Scholar
Steinberg, R.. Variations on a theme of Chevalley . Pacific J. Math., Vol. 9, No. 3, 875–891 (1959).CrossRefGoogle Scholar
Steinberg, R.. Endomorphisms of linear algebraic groups . Memoirs of the American Mathematical Society, No. 80 (1968).Google Scholar
Tits, J.. Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque . Journal für die reine und angewandte Mathematik 247 (1971): 196–220.Google Scholar
Tits, J.. Groups and group functors attached to Kac-Moody data. Arbeitstagung Bonn 1984, p. 193–223.CrossRefGoogle Scholar
Tits, J.. Immeubles de type affine. Buildings and the geometry of diagrams, Como (1984), L. A. Rosati, Lect. Notes in Math., Springer Verlag, Berlin 1986, 159–190.Google Scholar
Tits, J.. Uniqueness and presentation of Kac-Moody groups over fields . J. Algebra 105 (1987), no 2, 542–573.CrossRefGoogle Scholar
Tits, J.. Groupes associés aux algèbres de Kac-Moody. Astérisque, tome 177–178 (1989), Séminaire Bourbaki, exp. no 700, p. 7–31.Google Scholar
Tits, J.. Twin buildings and groups of Kac-Moody type, Groups, combinatorics and Geometry, Durham 1990, London Math. Soc. Lect. Series 165, Cambridge University Press (1992), 249–286.Google Scholar
Totaro, B.. Pseudo-abelian varieties . Annales scientifiques de l’École Normale Supérieure, serie 4, tome 46 (2013) no. 5, p. 693–721.CrossRefGoogle Scholar
Waterhouse, W. C. & Weisfeiler, B.. One-dimensional affine group schemes . J. Algebra, Vol. 66 (1980), no 2, p. 550–568.CrossRefGoogle Scholar
Yu, J.-K., Smooth models associated to concave functions in Bruhat-Tits theory. Autour des schémas en groupes, vol. III, 227–258, Panor. Synthèses, 47, Soc. Math. France, Paris (2015).Google Scholar
Zhu, X.. An introduction to affine Grassmannians and the geometric Satake equivalence. Geometry of moduli spaces and representation theory. IAS/Park City Math. Series 24 AMS (2017), 59–154.CrossRefGoogle Scholar
Zhu, X.. Affine Grassmannians and the geometric Satake in mixed characteristic . Ann. of Math. (2) 185 (2017), no 2, 403–492.CrossRefGoogle Scholar