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# Documents  Gille, Philippe | enregistrements trouvés : 8

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## Descent in Bruhat-Tits theory Prasad, Gopal | CIRM H

Post-edited

Algebraic and Complex Geometry;Lie Theory and Generalizations;Number Theory

Bruhat-Tits theory applies to a semisimple group G, defined over an henselian discretly valued field K, such that G admits a Borel K-subgroup after an extension of K. The construction of the theory goes then by a deep Galois descent argument for the building and also for the parahoric group scheme. In the case of unramified extension, that descent has been achieved by Bruhat-Tits at the end of [BT2]. The tamely ramified case is due to G. Rousseau [R]. Recently, G. Prasad found a new way to investigate the descent part of the theory. This is available in the preprints [Pr1, Pr2] dealing respectively with the unramified case and the tamely ramified case. It is much shorter and the method is based more on fine geometry of the building (e.g. galleries) than algebraic groups techniques. Bruhat-Tits theory applies to a semisimple group G, defined over an henselian discretly valued field K, such that G admits a Borel K-subgroup after an extension of K. The construction of the theory goes then by a deep Galois descent argument for the building and also for the parahoric group scheme. In the case of unramified extension, that descent has been achieved by Bruhat-Tits at the end of [BT2]. The tamely ramified case is due to G. ...

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## Autour des schémas en groupes.Ecole d'été "schémas en groupe" / Group schemes, a celebration of SGA3:Volume IIIMarseille # septembre 2011 Demazure, Michel ; Edixhoven, Bas ; Gille, Philippe ; Van der kallen, Wilberd ; Lee, Ting-Yu ; Pepin Lehalleur, Simon ; Romagny, Matthieu ; Tong, Jilong ; Yu, Jiu-Kang | Société Mathématique de France 2015

Congrès

- xx; 258 p.
ISBN 978-2-85629-820-6

Panoramas et synthèses , 0047

Localisation : Collection 1er étage

constante de structure # cross-section de Steinberg # donnée radicielle # isotropie # loi de groupe birationnelle # modèle de Néron # modèle # quotient adjoint # réductibilité # règle des signes # représentation fondamentale # schéma en groupes réductifs # schéma p-polynomial # schéma en groupes de Chevalley # schéma en groupes unipotent # schéma en groupes # sous-groupe parabolique # système de racines

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## Bruhat-Tits theory of quasi-split groups Rémy, Bertrand | CIRM H

Multi angle

Algebra

The goal of this lecture is to present the construction of the Bruhat-Tits buildings attached to a quasi-split (that is admitting a Borel subgroup) semisimple group G defined over an henselian discretly valued field K and also the construction of the parahoric group schemes parametrized by the points of the buildings. The building part is [BT1] and the group scheme part corresponds to the four first sections of [BT2] but could also be treated by Yu's method [Y] namely by using Raynaud's theory of group schemes [BLR]. The goal of this lecture is to present the construction of the Bruhat-Tits buildings attached to a quasi-split (that is admitting a Borel subgroup) semisimple group G defined over an henselian discretly valued field K and also the construction of the parahoric group schemes parametrized by the points of the buildings. The building part is [BT1] and the group scheme part corresponds to the four first sections of [BT2] but could also be treated by ...

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## Basics on affine Grassmanianns Richarz, Timo | CIRM H

Multi angle

Algebraic and Complex Geometry;Number Theory

The aim is to give an introduction to the basic theory of affine Grassmannians and affine flag varieties. We put special emphasis on the utility of dynamic methods in sense of Drinfeld [D], and the utility of non-constant group schemes. We plan to adress the following aspects:
• Affine Grassmannians as moduli spaces of G-bundles, and as quotients of loop groups ;
• Cell decompositions of affine Grassmannians and affine flag varieties via dynamic methods: Iwahori, Cartan and Iwasawa decompositions ;
• Schubert varieties, Demazure resolutions, Convolution morphisms, Combinatorial structures ;
• Moduli spaces of G-bundles with level structure versus bundles under non-constant group schemes ;
• Beilinson-Drinfeld type deformations of affine Grassmannians ;
• Relation to the local geometry of moduli spaces of Drinfeld shtukas and Shimura varieties.
The aim is to give an introduction to the basic theory of affine Grassmannians and affine flag varieties. We put special emphasis on the utility of dynamic methods in sense of Drinfeld [D], and the utility of non-constant group schemes. We plan to adress the following aspects:
• Affine Grassmannians as moduli spaces of G-bundles, and as quotients of loop groups ;
• Cell decompositions of affine Grassmannians and affine flag varieties via dynamic ...

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## Point de vue de Berkovich sur l’immeuble et filtrations Mayeux, Arnaud | CIRM H

Multi angle

L’immeuble réduit de Bruhat-Tits de G (réductif connexe) se plonge dans l’analytifié $G^{an}$. Cela est dû à Berkovich et Rémy-Thuillier-Werner. Nous expliquerons cela puis nous expliquerons que l’on peut définir naturellement dans ce cadre des filtrations analytiques dont les points rationnels coïncident dans certains cas avec les groupes de Moy-Prasad.

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## Twisted Kac-Moody groups over the integers Lourenço, João | CIRM H

Multi angle

Algebraic and Complex Geometry;Number Theory

In geometric representation theory, one is interested in studying the geometry of affine Grassmannians of quasi-split simply-connected reductive groups. In this endeavor, one of the main techniques, introduced by Faltings in the split case, consists in constructing natural realisations of these ind-schemes over the integers. In the twisted case, this was done by Pappas and Rapoport in the tamely ramified case, i.e. over $\mathbb{Z}[1/e]$, where $e = 2$ or $3$ is the order of the automorphism group of the split form we are dealing with. We explain how to extend the parahoric group scheme that appeared in work of Pappas, Rapoport, Tits and Zhu to the polynomial ring $\mathbb{Z}[t]$ with integer coefficients and additionally how the group scheme obtained in char. $e$ can be regarded as a parahoric model of a basic exotic pseudo-reductive group. Then we study the geometry of the affine Grassmannian and also its global deformation à la Beilinson-Drinfeld, recovering all the known results in the literature away from $e = 0$. This also has some pertinence to the study of local models of Shimura varieties in wildly ramified cases. In geometric representation theory, one is interested in studying the geometry of affine Grassmannians of quasi-split simply-connected reductive groups. In this endeavor, one of the main techniques, introduced by Faltings in the split case, consists in constructing natural realisations of these ind-schemes over the integers. In the twisted case, this was done by Pappas and Rapoport in the tamely ramified case, i.e. over \$\math...

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## Central simple algebras and Galois cohomology Gille, Philippe ; Szamuely, Tamas | Cambridge University Press 2006

Ouvrage

- 343 p.
ISBN 978-0-521-86103-8

Cambridge studies in advanced mathematics , 0101

Localisation : Ouvrage RdC (GILL)

cohomologie galoisienne # groupe de Brauer # k-théorie de Milnor # algèbre centrale semi-simple

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## Torsors, reductive group schemes and extended affine Lie algebras Gille, Philippe ; Pianzola, Arturo | American Mathematical Society 2013

Ouvrage

- v; 112 p.
ISBN 978-0-8218-8774-5

Memoirs of the american mathematical society , 1063

Localisation : Collection 1er étage

Algèbres de Kac-Moody # groupes algébriques linéaires # géométrie algébrique

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