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].[-]

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.[-]

Let $K$ be a discretely valued field with ring of integers $R$ and let $d$ be a positive integer. Then the rank $d$ free $R$-submodules of $K^{d}$ (called $R$-lattices) are the $0$-simplices of an infinite simplicial complex called a Bruhat-Tits building. If $O$ is an order in the ring of $d\times d$ matrices over $K$, then the collection of lattices that are also $O$-modules (called $O$-lattices) is a non-empty, bounded and convex subset of the building. Determining what these subsets are is in general a difficult question.
I will report on joint work with Yassine El Maazouz, Gabriele Nebe, Marvin Hahn, and Bernd Sturmfels describing the geometric features of the set of $O$-lattices for some particular orders. If time permits, I will also define spherical codes in Bruhat-Tits buildings and show how these fit in this framework and how they give rise to codes of submodules over chain rings.[-]