An illustrated introduction to buildings

by Paul Philippe (Institut Camille Jordan)

Salle Mirzhakani (IHP, Paris)

Salle Mirzhakani

IHP, Paris


In order to study reductive groups (such as $SL_n$) over arbitrary fields, Jacques Tits introduced a simplicial analog of symmetric spaces: buildings. A building is a simplicial complex obtained by gluing, in a regular fashion (along walls), several copies of a model complex (its apartments). A reductive group $G$ over a field $k$ admits a transitive action on a building: its spherical building. From the geometry and the combinatorics of the building we can deduce many properties of $G(k)$. If the base field is discretely valued, taking profit from the valuation François Bruhat and J. Tits have constructed a bigger building on which $G(k)$ acts, its Bruhat-Tits building. This has many applications in representation theory of p-adic groups, as well as in the study of arithmetic groups.

In this talk I will introduce abstract buildings and give the example of the spherical and the Bruhat-Tits buildings of $SL_n$. In particular I will illustrate Nagao's theorem, which gives an explicit expression of $SL_2(k[t])$ as an amalgamated product.