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 OK–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].

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.
​Gopal Prasad is Professor in University of Michigan. His research topic is algebraic groups and arithmetic groups.

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 :
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• 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. 

Timo Richarz is Assistant Professor at TU Darmstadt, Germany. His research area is algebraic geometry and the Langlands program.

One of the most important applications of the affine Grassmannian is the geometric Satake equivalence which claims that, if G is a complex connected reductive group, the Satake category of G, i.e. the category of G(C[[t]])– equivariant perverse sheaves on the affine Grassmannian of G, with coefficients in a commutative Noetherian ring k of finite global dimension, is equivalent to the category of finitely generated representations of the split connected reductive group over k which is Langlands dual to G. This equivalence is important because on one hand it gives a geometric description of the category of representations of any reductive group, and on the other hand because it gives a “concrete” construction of the Langlands dual group in terms of the original group. These lectures will provide an overview of the proof of this equivalence by Mirković-Vilonen [MV], and describe recent developments related to the description of the tilting objects in the Satake category. These lectures will be based in particular on [BR].

​Simon Riche is Professor at University of Clermont-Ferrand Auvergne, his research area is representation theory.

The geometric Satake equivalence can be regarded as a categorification of the classical Satake isomorphism. In this lecture series, I will discuss a general decategorification process, known as the categorical trace construction, and apply this construction to the geometric Satake equivalence. Shtukas appear naturally in the construction, and surprisingly, this decategorification process gives more information than the classical Satake isomorphism and has significant arithmetic applications to the classical Langlands program. Time permitting, I may also discuss this construction for Bezrukavnikov’s equivalence.​

[BLR] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete 21 (1990), Springer.

[BT1] F. Bruhat, J. Tits, Groupes réductifs sur un corps local. I. Données radicielles valuées, Inst. Hautes Etudes Sci. Publ. Math. 41 (1972), 5–251.

[BT2] F. Bruhat, J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Etudes Sci. Publ. Math. 60 (1984), 197–376.

[D] V. G. Drinfeld : On algebraic spaces with an action of Gm, preprint (2013).

[F] R. Fedorov, Affine Grassmannians of group schemes and exotic principal bundles over A 1 , Amer. J. Math. 138 (2016), 879-906.

[Ga] D. Gaitsgory, Progrès récents dans la théorie de Langlands géométrique, Séminaire Bourbaki, Exp. No. 1109, vol. 2015/2016, Astérisque No. 390 (2017), 139-168.

[MV] I. Mirković, K. Vilonen, Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), 13-24.

[Pr1] G. Prasad, A new approach to unramified descent in Bruhat-Tits theory.

[Pr2] G. Prasad, Finite group actions on reductive groups and buildings and tamely-ramified descent in Bruhat-Tits theory, 

[R] G. Rousseau, Immeubles des groupes réductifs sur les corps locaux Publications mathématiques d’Orsay 130 (1977). 

[Y] J.-K. Yu, 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.