View source for Bruhat-Tits building

A building (cf. also [[Tits building|Tits building]]) which is defined for a connected [[Reductive group|reductive group]] over a [[Field|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b1109601.png" /> be a [[Field|field]] which is complete with respect to the non-trivial valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b1109602.png" /> and has a perfect residue class field. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b1109603.png" /> be a connected, reductive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b1109604.png" />-group. First, assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b1109605.png" /> to be semi-simple (cf. [[Semi-simple group|Semi-simple group]]).

==Apartments.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b1109606.png" /> be a maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b1109607.png" />-split [[Torus|torus]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b1109608.png" /> and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b1109609.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096010.png" />) the [[Centralizer|centralizer]] (respectively, normalizer; cf. [[Normalizer of a subset|Normalizer of a subset]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096013.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096014.png" />) denote the group of cocharacters (respectively, characters) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096015.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096016.png" /> be the canonical perfect pairing. Then there is a unique group [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096019.png" /> (i.e., the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096020.png" />-rational characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096021.png" />). One can show that there is a unique affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096022.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096023.png" /> together with a group homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096024.png" /> (i.e., the affine bijections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096025.png" />) extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096026.png" />, called the (empty) apartment associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096027.png" />.

==Filtrations of the root subgroups.==
Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096028.png" /> the [[Root system|root system]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096029.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096030.png" /> and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096031.png" />, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096032.png" /> the root subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096033.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096034.png" />. Then, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096035.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096036.png" /> contains exactly one element, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096037.png" />. An affine mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096038.png" /> is called an affine root if the vector part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096040.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096041.png" /> and if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096042.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096043.png" />. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096044.png" /> is abbreviated as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096045.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096047.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096048.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096049.png" /> be the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096050.png" /> generated by all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096051.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096052.png" />.

==Simplicial structures.==
Two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096053.png" /> are called equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096055.png" /> have the same sign or are both equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096056.png" /> for all affine roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096057.png" />. One obtains a (poly-) simplicial complex (i.e., a direct product of simplicial complexes) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096058.png" /> by defining the faces to be the equivalence classes.

==Building.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096059.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096060.png" /> if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096061.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096063.png" />. There is a canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096064.png" />-action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096065.png" /> induced by left-multiplication on the first factor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096066.png" />. One can identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096067.png" /> with its canonical image in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096068.png" />. The subsets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096069.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096070.png" />, are called apartments and the subsets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096071.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096073.png" /> a face in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096074.png" />, are called faces. One can equip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096075.png" /> with a [[Metric|metric]] which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096076.png" />-invariant. This metric coincides on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096077.png" /> with the metric induced by the scalar product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096078.png" /> which is invariant under the Weyl group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096079.png" />. The metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096080.png" /> together with these structures is called the Bruhat–Tits building of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096081.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096082.png" /> is not semi-simple, the Bruhat–Tits building of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096083.png" /> is, by definition, the Bruhat–Tits building of the derived group (cf. [[Commutator subgroup|Commutator subgroup]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096084.png" />.

===Example.===
Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096085.png" />, and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096087.png" /> the valuation ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096088.png" /> and a uniformizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096089.png" />, respectively. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096091.png" />-lattice is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096092.png" />-submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096093.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096094.png" />. Then the Bruhat–Tits building of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096095.png" /> is the topological realization of the following simplicial complex: the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096096.png" />-simplices are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096097.png" />-lattices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096098.png" /> up to homothety and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b11096099.png" />-simplices are pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b110960100.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b110960101.png" />-lattices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b110960102.png" /> up to homothety with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110960/b110960103.png" />.

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

Originally, the Bruhat–Tits building was the essential technical tool for the classification of reductive groups over local fields (cf. [[Reductive group|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Bruhat,   J. Tits,   "Groupes réductifs sur un corps local I, II"  ''IHES Publ. Math.'' , '''41,50'''  (1972–1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Landvogt,   "A compactification of the Bruhat–Tits building" , ''Lecture Notes in Mathematics'' , '''1619''' , Springer  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Tits,   "Reductive groups over local fields" , ''Proc. Symp. Pure Math.'' , '''33''' , Amer. Math. Soc.  (1979)  pp. 29–69</TD></TR></table>