Trends in Mathematics

Abstract. A Bruhat-Tits building of a p-adic reductive group is a p-adic analogue of the global Riemannian symmetric space of a semisimple Lie group. It is useful in studying structure of G and its various compact open subgroups as well as its representations. This note is a short introduction to the theory starting from the Tits system. Let G be a semisimple Lie group and K be a maximal compact subgroup of G which is the set of fixed poiints under a Cartan involution. Then G/K is a global Riemannian symmetric space. For example, SL2(R)/SO2(R) is the Poincare ́ upper half plane. When G is a semisimple p-adic group, its maximal compact subgroup K is always open (for example, SLn(Zp) ⊂ SLn(Qp)) hence the coset space G/K is a discrete set. The Bruhat-Tits building B of G is a complete metric space having a (poly)simplicial complex structure with isometric and simplicial G-action. The set G/K is a subset of vertices of B and the G-action on these vertices is the same as the translations on G/K. This note is a short introduction to Bruhat-Tits buildings of p-adic reductive groups, which were defined and throughly investigated in [1, 2]