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 $ K $ be a field which is complete with respect to the non-trivial valuation $ \omega : {K ^ \times } \rightarrow \mathbf Z $ and has a perfect residue class field. Let $ G $ be a connected, reductive $ K $- group. First, assume $ G $ to be semi-simple (cf. Semi-simple group).

Apartments.

Let $ S $ be a maximal $ K $- split torus in $ G $ and denote by $ Z $( respectively, $ N $) the centralizer (respectively, normalizer; cf. Normalizer of a subset) of $ S $ in $ G $. Let $ X _ {*} ( S ) $( respectively, $ X ^ {*} ( S ) $) denote the group of cocharacters (respectively, characters) of $ S $ and let $ {\langle {\cdot, \cdot } \rangle } : {X ^ {*} ( S ) \times X _ {*} ( S ) } \rightarrow \mathbf Z $ be the canonical perfect pairing. Then there is a unique group homomorphism $ \nu : {Z ( K ) } \rightarrow {V = X _ {*} ( S ) \otimes _ {\mathbf Z} \mathbf R } $ such that $ \langle {\chi, \nu ( z ) } \rangle = - \omega ( \chi ( z ) ) $ for all $ \chi \in X ^ {*} _ {K} ( Z ) $( i.e., the group of $ K $- rational characters of $ Z $). One can show that there is a unique affine $ V $- space $ A $ together with a group homomorphism $ \nu : {N ( K ) } \rightarrow { { \mathop{\rm Aff} } ( A ) } $( i.e., the affine bijections $ A \rightarrow A $) extending $ \nu : {Z ( K ) } \rightarrow {V \subset { \mathop{\rm Aff} } ( A ) } $, called the (empty) apartment associated with $ S $.

Filtrations of the root subgroups.

Denote by $ \Phi $ the root system of $ G $ with respect to $ S $ and, for $ a \in \Phi $, by $ U _ {a} $ the root subgroup of $ G $ associated with $ a $. Then, for $ u \in U _ {a} ( K ) \backslash \{ 1 \} $, the set $ U _ {- a } ( K ) uU _ {- a } ( K ) \cap N ( K ) $ contains exactly one element, denoted by $ m ( u ) $. An affine mapping $ \alpha : A \rightarrow \mathbf R $ is called an affine root if the vector part $ a $ of $ \alpha $ is contained in $ \Phi $ and if there exists a $ u \in U _ {a} ( K ) \backslash \{ 1 \} $ such that $ \alpha ^ {-1 } ( 0 ) = \{ {x \in A } : {\nu ( m ( u ) ) ( x ) = x } \} $. In that case $ \alpha $ is abbreviated as $ \alpha ( a,u ) $. For $ x \in A $ and $ a \in \Phi $, let $ U _ {a,x } = \{ {u \in U _ {a} ( K ) } : {\alpha ( a,u ) ( x ) \geq 0 } \} \cup \{ 1 \} $ and let $ U _ {x} $ be the subgroup of $ G ( K ) $ generated by all $ U _ {a,x } $ for $ a \in \Phi $.

Simplicial structures.

Two points $ x,y \in A $ are called equivalent if $ \alpha ( x ) $ and $ \alpha ( y ) $ have the same sign or are both equal to $ 0 $ for all affine roots $ \alpha $. One obtains a (poly-) simplicial complex (i.e., a direct product of simplicial complexes) in $ A $ by defining the faces to be the equivalence classes.

Building.

Let $ { \mathop{\rm BT} } ( G,K ) = G ( K ) \times A/ \sim $, where $ ( g,x ) \sim ( h,y ) $ if there exists an $ n \in N ( K ) $ such that $ \nu ( n ) ( x ) = y $ and $ g ^ {- 1 } hn \in U _ {x} $. There is a canonical $ G ( K ) $- action on $ { \mathop{\rm BT} } ( G,K ) $ induced by left-multiplication on the first factor of $ G ( K ) \times A $. One can identify $ A $ with its canonical image in $ { \mathop{\rm BT} } ( G,K ) $. The subsets of the form $ gA $, for $ g \in G ( K ) $, are called apartments and the subsets of the form $ gF $, for $ g \in G ( K ) $ and $ F $ a face in $ A $, are called faces. One can equip $ { \mathop{\rm BT} } ( G,K ) $ with a metric which is $ G ( K ) $- invariant. This metric coincides on $ A $ with the metric induced by the scalar product on $ V $ which is invariant under the Weyl group of $ \Phi $. The metric space $ { \mathop{\rm BT} } ( G,K ) $ together with these structures is called the Bruhat–Tits building of $ G $.

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

Example.

Assume $ G = { \mathop{\rm SL} } _ {2} $, and denote by $ {\mathcal O} $ and $ \pi $ the valuation ring of $ K $ and a uniformizer of $ {\mathcal O} $, respectively. An $ {\mathcal O} $- lattice is a free $ {\mathcal O} $- submodule of $ K ^ {2} $ of rank $ 2 $. Then the Bruhat–Tits building of $ { \mathop{\rm SL} } _ {2} $ is the topological realization of the following simplicial complex: the $ 0 $- simplices are the $ {\mathcal O} $- lattices in $ K ^ {2} $ up to homothety and the $ 1 $- simplices are pairs $ L \subset L ^ \prime $ of $ {\mathcal O} $- lattices in $ K ^ {2} $ up to homothety with $ \pi L ^ \prime \subset L $.

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