Bruhat-Tits building

A building (cf. also Tits building) which is defined for a connected reductive group over a field which is complete with respect to a non-trivial discrete valuation and has a perfect residue class field. It is the topological realization of the (poly-) simplicial complex of all parahoric subgroups. One way to define it more precisely is as follows.

Let be a field which is complete with respect to the non-trivial valuation and has a perfect residue class field. Let be a connected, reductive -group. First, assume to be semi-simple (cf. Semi-simple group).

Apartments.

Let be a maximal -split torus in and denote by (respectively, ) the centralizer (respectively, normalizer; cf. Normalizer of a subset) of in . Let (respectively, ) denote the group of cocharacters (respectively, characters) of and let be the canonical perfect pairing. Then there is a unique group homomorphism such that for all (i.e., the group of -rational characters of ). One can show that there is a unique affine -space together with a group homomorphism (i.e., the affine bijections ) extending , called the (empty) apartment associated with .

Filtrations of the root subgroups.

Denote by the root system of with respect to and, for , by the root subgroup of associated with . Then, for , the set contains exactly one element, denoted by . An affine mapping is called an affine root if the vector part of is contained in and if there exists a such that . In that case is abbreviated as . For and , let and let be the subgroup of generated by all for .

Simplicial structures.

Two points are called equivalent if and have the same sign or are both equal to for all affine roots . One obtains a (poly-) simplicial complex (i.e., a direct product of simplicial complexes) in by defining the faces to be the equivalence classes.

Building.

Let , where if there exists an such that and . There is a canonical -action on induced by left-multiplication on the first factor of . One can identify with its canonical image in . The subsets of the form , for , are called apartments and the subsets of the form , for and a face in , are called faces. One can equip with a metric which is -invariant. This metric coincides on with the metric induced by the scalar product on which is invariant under the Weyl group of . The metric space together with these structures is called the Bruhat–Tits building of .

If is not semi-simple, the Bruhat–Tits building of is, by definition, the Bruhat–Tits building of the derived group (cf. Commutator subgroup) of .

Example.

Assume , and denote by and the valuation ring of and a uniformizer of , respectively. An -lattice is a free -submodule of of rank . Then the Bruhat–Tits building of is the topological realization of the following simplicial complex: the -simplices are the -lattices in up to homothety and the -simplices are pairs of -lattices in up to homothety with .

General references for Bruhat–Tits buildings are [a1] and [a2]. (The situations considered there are more general.) A good overview can be found in [a3].

Originally, the Bruhat–Tits building was the essential technical tool for the classification of reductive groups over local fields (cf. Reductive group). There are further applications, e.g. in the representation theory of reductive groups over local fields, in the theory of p-adic symmetric spaces and in the theory of Shimura varieties.

References

[a1]	F. Bruhat, J. Tits, "Groupes réductifs sur un corps local I, II" IHES Publ. Math. , 41,50 (1972–1984)
[a2]	E. Landvogt, "A compactification of the Bruhat–Tits building" , Lecture Notes in Mathematics , 1619 , Springer (1996)
[a3]	J. Tits, "Reductive groups over local fields" , Proc. Symp. Pure Math. , 33 , Amer. Math. Soc. (1979) pp. 29–69