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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.
 
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]]).
+
Let $  K $
 +
be a [[Field|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|Semi-simple group]]).
  
 
==Apartments.==
 
==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" />.
+
Let $  S $
 +
be a maximal $  K $-
 +
split [[Torus|torus]] in $  G $
 +
and denote by $  Z $(
 +
respectively, $  N $)  
 +
the [[Centralizer|centralizer]] (respectively, normalizer; cf. [[Normalizer of a subset|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|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.==
 
==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" />.
+
Denote by $  \Phi $
 +
the [[Root system|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.==
 
==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.
+
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.==
 
==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" />.
+
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|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 <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" />.
+
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|Commutator subgroup]]) of $  G $.
  
 
===Example.===
 
===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" />.
+
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 [[#References|[a1]]] and [[#References|[a2]]]. (The situations considered there are more general.) A good overview can be found in [[#References|[a3]]].
 
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]]].

Latest revision as of 06:29, 30 May 2020


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
How to Cite This Entry:
Bruhat-Tits building. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bruhat-Tits_building&oldid=14190
This article was adapted from an original article by E. Landvogt (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article