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''of a connected reductive algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810301.png" /> defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810302.png" />''
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{{TEX|done}}
 +
''of a connected reductive algebraic group $  G $
 +
defined over a field $  k $ ''
  
A system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810303.png" /> of non-zero weights of the adjoint representation of a maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810304.png" />-split torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810305.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810306.png" /> in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810307.png" /> of this group (cf. [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]). The weights themselves are called the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810308.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810309.png" />. The relative root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103010.png" />, which can be seen as a subset of its linear envelope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103011.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103013.png" /> is the group of rational characters of the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103014.png" />, is a [[Root system|root system]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103015.png" /> be the normalizer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103016.png" /> the centralizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103018.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103019.png" /> is the connected component of the unit of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103020.png" />; the finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103021.png" /> is called the Weyl group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103022.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103023.png" />, or the relative Weyl group. The adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103025.png" /> defines a linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103027.png" />. This representation is faithful and its image is the [[Weyl group|Weyl group]] of the root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103028.png" />, which enables one to identify these two groups. Since two maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103029.png" />-split tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103032.png" /> are conjugate over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103033.png" />, the relative root systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103034.png" /> and the relative Weyl groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103036.png" />, are isomorphic, respectively. Hence they are often denoted simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103038.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103039.png" /> is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103040.png" />, the relative root system and the relative Weyl group coincide, respectively, with the usual (absolute) root system and Weyl group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103041.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103042.png" /> be the weight subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103043.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103044.png" />, corresponding to the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103045.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103046.png" /> is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103047.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103048.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103049.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103050.png" /> is a reduced root system; this is not so in general: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103051.png" /> does not have to be reduced and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103052.png" /> can be greater than 1. The relative root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103053.png" /> is irreducible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103054.png" /> is simple over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103055.png" />.
 
  
The relative root system plays an important role in the description of the structure and in the classification of semi-simple algebraic groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103056.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103057.png" /> be semi-simple, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103058.png" /> be a maximal torus defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103059.png" /> and containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103060.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103062.png" /> be the groups of rational characters of the tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103064.png" /> with fixed compatible order relations, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103065.png" /> be a corresponding system of simple roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103066.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103067.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103068.png" /> be the subsystem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103069.png" /> consisting of the characters which are trivial on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103070.png" />. Moreover, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103071.png" /> be the system of simple roots in the relative root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103072.png" /> defined by the order relation chosen on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103073.png" />; it consists of the restrictions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103074.png" /> of the characters of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103075.png" />. The Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103076.png" /> acts naturally on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103077.png" />, and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103078.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103080.png" />-index of the semi-simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103081.png" />. The role of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103082.png" />-index is explained by the following theorem: Every semi-simple group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103083.png" /> is uniquely defined, up to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103084.png" />-isomorphism, by its class relative to an isomorphism over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103085.png" />, its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103086.png" />-index and its [[Anisotropic kernel|anisotropic kernel]]. The relative root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103087.png" /> is completely defined by the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103088.png" /> and by the set of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103090.png" /> (equal to 1 or 2), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103091.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103092.png" />. Conversely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103095.png" />, can be determined from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103096.png" />-index. In particular, two elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103097.png" /> have one and the same restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103098.png" /> if and only if they are located in the same orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103099.png" />; this defines a bijection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030100.png" /> and the set of orbits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030101.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030102.png" />.
+
A system $  \Phi _{k} (S,\  G) $
 +
of non-zero weights of the adjoint representation of a maximal  $  k $ -
 +
split torus  $  S $
 +
of the group  $  G $
 +
in the Lie algebra  $  \mathfrak g $
 +
of this group (cf. [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]). The weights themselves are called the roots of $  G $
 +
relative to $  S $ .  
 +
The relative root system  $  \Phi _{k} (S,\  G) $ ,  
 +
which can be seen as a subset of its linear envelope  $  L $
 +
in the space  $  X(S) \otimes _ {\mathbf Z} \mathbf R $ ,
 +
where  $  X(S) $
 +
is the group of rational characters of the torus  $  S $ ,  
 +
is a [[Root system|root system]]. Let  $  N(S) $
 +
be the normalizer and  $  Z(S) $
 +
the centralizer of $  S $
 +
in $  G $ .  
 +
Then  $  Z(S) $
 +
is the connected component of the unit of the group  $  N(S) $ ;
 +
the finite group $  W _{k} (S,\  G) = N(S)/Z(S) $
 +
is called the Weyl group of $  G $
 +
over  $  k $ ,
 +
or the relative Weyl group. The adjoint representation of $  N(S) $
 +
in  $  \mathfrak g $
 +
defines a linear representation of  $  W _{k} (S,\  G) $
 +
in  $  L $ .  
 +
This representation is faithful and its image is the [[Weyl group|Weyl group]] of the root system $  \Phi _{k} (S,\  G) $ ,
 +
which enables one to identify these two groups. Since two maximal  $  k $ -
 +
split tori  $  S _{1} $
 +
and  $  S _{2} $
 +
in  $  G $
 +
are conjugate over  $  k $ ,
 +
the relative root systems  $  \Phi _{k} (S _{i} ,\  G) $
 +
and the relative Weyl groups  $  W _{k} (S _{i} ,\  G) $ ,  
 +
$  i=1,\  2 $ ,
 +
are isomorphic, respectively. Hence they are often denoted simply by  $  \Phi _{k} (G) $
 +
and  $  W _{k} (G) $ .  
 +
When  $  G $
 +
is split over  $  k $ ,  
 +
the relative root system and the relative Weyl group coincide, respectively, with the usual (absolute) root system and Weyl group of  $  G $ .  
 +
Let  $  g _ \alpha  $
 +
be the weight subspace in  $  \mathfrak g $
 +
relative to  $  S $ ,  
 +
corresponding to the root  $  \alpha \in \Phi _{k} (S,\  G) $ .  
 +
If  $  G $
 +
is split over  $  k $ ,  
 +
then  $  \mathop{\rm dim}\nolimits \  g _ \alpha  = 1 $
 +
for any  $  \alpha $ ,
 +
and $  \Phi _{k} (G) $
 +
is a reduced root system; this is not so in general: $  \Phi _{k} (G) $
 +
does not have to be reduced and $  \mathop{\rm dim}\nolimits \  g _ \alpha  $
 +
can be greater than 1. The relative root system  $  \Phi _{k} (G) $
 +
is irreducible if  $  G $
 +
is simple over  $  k $ .
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030103.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030104.png" /> is the corresponding orbit, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030105.png" /> is any connected component in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030106.png" /> not all vertices of which lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030107.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030108.png" /> is the sum of the coefficients of the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030109.png" /> in the decomposition of the highest root of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030110.png" /> in simple roots.
 
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030112.png" />, then the above relative root system and relative Weyl group are naturally identified with the root system and Weyl group, respectively, of the corresponding symmetric space.
+
The relative root system plays an important role in the description of the structure and in the classification of semi-simple algebraic groups over  $  k $ .
 +
Let  $  G $
 +
be semi-simple, and let  $  T $
 +
be a maximal torus defined over  $  k $
 +
and containing  $  S $ .
 +
Let  $  X(S) $
 +
and  $  X(T) $
 +
be the groups of rational characters of the tori  $  S $
 +
and  $  T $
 +
with fixed compatible order relations, let  $  \Delta $
 +
be a corresponding system of simple roots of  $  G $
 +
relative to  $  T $ ,
 +
and let  $  \Delta _{0} $
 +
be the subsystem in  $  \Delta $
 +
consisting of the characters which are trivial on  $  S $ .
 +
Moreover, let  $  \Delta _{k} $
 +
be the system of simple roots in the relative root system  $  \Phi _{k} (S,\  G) $
 +
defined by the order relation chosen on  $  X(S) $ ;
 +
it consists of the restrictions to  $  S $
 +
of the characters of the system  $  \Delta $ .
 +
The Galois group  $  \Gamma = \mathop{\rm Gal}\nolimits (k _{s} /k) $
 +
acts naturally on  $  \Delta $ ,
 +
and the set  $  \{ \Delta ,\  \Delta _{0} ,  \textrm{ the  action  of }  \Gamma  \textrm{ on }  \Delta \} $
 +
is called the  $  k $ -
 +
index of the semi-simple group  $  G $ .
 +
The role of the  $  k $ -
 +
index is explained by the following theorem: Every semi-simple group over  $  k $
 +
is uniquely defined, up to a  $  k $ -
 +
isomorphism, by its class relative to an isomorphism over  $  k _{s} $ ,
 +
its  $  k $ -
 +
index and its [[Anisotropic kernel|anisotropic kernel]]. The relative root system  $  \Phi _{k} (G) $
 +
is completely defined by the system  $  \Delta _{k} $
 +
and by the set of natural numbers  $  n _ \alpha  $ ,
 +
$  \alpha \in \Delta _{k} $ (
 +
equal to 1 or 2), such that  $  n _ \alpha  \alpha \in \Phi _{k} (G) $
 +
but  $  (n _ \alpha  + 1) \alpha \notin \Phi _{k} (G) $ .  
 +
Conversely,  $  \Delta _{k} $
 +
and  $  n _ \alpha  $ ,
 +
$  \alpha \in \Delta _{k} $ ,
 +
can be determined from the  $  k $ -
 +
index. In particular, two elements from  $  \Delta \setminus \Delta _{0} $
 +
have one and the same restriction to  $  S $
 +
if and only if they are located in the same orbit of  $  \Gamma $ ;
 +
this defines a bijection between  $  \Delta _{k} $
 +
and the set of orbits of  $  \Gamma $
 +
into  $  \Delta \setminus \Delta _{0} $ .
 +
 
 +
 
 +
If  $  \gamma \in \Delta _{k} $ ,
 +
if  $  O _ \gamma  \subset \Delta \setminus \Delta _{0} $
 +
is the corresponding orbit, if  $  \Delta ( \gamma ) $
 +
is any connected component in  $  \Delta _{0} \cup O _ \gamma  $
 +
not all vertices of which lie in  $  \Delta _{0} $ ,
 +
then  $  n _ \gamma  $
 +
is the sum of the coefficients of the roots  $  \alpha \in \Delta ( \gamma ) \cap O _ \gamma  $
 +
in the decomposition of the highest root of the system  $  \Delta ( \gamma ) $
 +
in simple roots.
 +
 
 +
If  $  k = \mathbf R $ ,
 +
$  \overline{k}  = \mathbf C $ ,  
 +
then the above relative root system and relative Weyl group are naturally identified with the root system and Weyl group, respectively, of the corresponding symmetric space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Tits, "Sur la classification des groupes algébriques semi-simples" ''C.R. Acad. Sci. Paris'' , '''249''' (1959) pp. 1438–1440 {{MR|0106967}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Tits, "Classification of algebraic simple groups" , ''Algebraic Groups and Discontinuous Subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc. (1966) pp. 33–62 {{MR|}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Tits, "Sur la classification des groupes algébriques semi-simples" ''C.R. Acad. Sci. Paris'' , '''249''' (1959) pp. 1438–1440 {{MR|0106967}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Tits, "Classification of algebraic simple groups" , ''Algebraic Groups and Discontinuous Subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc. (1966) pp. 33–62 {{MR|}} {{ZBL|}} </TD></TR></table>

Latest revision as of 17:30, 17 December 2019

of a connected reductive algebraic group $ G $ defined over a field $ k $


A system $ \Phi _{k} (S,\ G) $ of non-zero weights of the adjoint representation of a maximal $ k $ - split torus $ S $ of the group $ G $ in the Lie algebra $ \mathfrak g $ of this group (cf. Weight of a representation of a Lie algebra). The weights themselves are called the roots of $ G $ relative to $ S $ . The relative root system $ \Phi _{k} (S,\ G) $ , which can be seen as a subset of its linear envelope $ L $ in the space $ X(S) \otimes _ {\mathbf Z} \mathbf R $ , where $ X(S) $ is the group of rational characters of the torus $ S $ , is a root system. Let $ N(S) $ be the normalizer and $ Z(S) $ the centralizer of $ S $ in $ G $ . Then $ Z(S) $ is the connected component of the unit of the group $ N(S) $ ; the finite group $ W _{k} (S,\ G) = N(S)/Z(S) $ is called the Weyl group of $ G $ over $ k $ , or the relative Weyl group. The adjoint representation of $ N(S) $ in $ \mathfrak g $ defines a linear representation of $ W _{k} (S,\ G) $ in $ L $ . This representation is faithful and its image is the Weyl group of the root system $ \Phi _{k} (S,\ G) $ , which enables one to identify these two groups. Since two maximal $ k $ - split tori $ S _{1} $ and $ S _{2} $ in $ G $ are conjugate over $ k $ , the relative root systems $ \Phi _{k} (S _{i} ,\ G) $ and the relative Weyl groups $ W _{k} (S _{i} ,\ G) $ , $ i=1,\ 2 $ , are isomorphic, respectively. Hence they are often denoted simply by $ \Phi _{k} (G) $ and $ W _{k} (G) $ . When $ G $ is split over $ k $ , the relative root system and the relative Weyl group coincide, respectively, with the usual (absolute) root system and Weyl group of $ G $ . Let $ g _ \alpha $ be the weight subspace in $ \mathfrak g $ relative to $ S $ , corresponding to the root $ \alpha \in \Phi _{k} (S,\ G) $ . If $ G $ is split over $ k $ , then $ \mathop{\rm dim}\nolimits \ g _ \alpha = 1 $ for any $ \alpha $ , and $ \Phi _{k} (G) $ is a reduced root system; this is not so in general: $ \Phi _{k} (G) $ does not have to be reduced and $ \mathop{\rm dim}\nolimits \ g _ \alpha $ can be greater than 1. The relative root system $ \Phi _{k} (G) $ is irreducible if $ G $ is simple over $ k $ .


The relative root system plays an important role in the description of the structure and in the classification of semi-simple algebraic groups over $ k $ . Let $ G $ be semi-simple, and let $ T $ be a maximal torus defined over $ k $ and containing $ S $ . Let $ X(S) $ and $ X(T) $ be the groups of rational characters of the tori $ S $ and $ T $ with fixed compatible order relations, let $ \Delta $ be a corresponding system of simple roots of $ G $ relative to $ T $ , and let $ \Delta _{0} $ be the subsystem in $ \Delta $ consisting of the characters which are trivial on $ S $ . Moreover, let $ \Delta _{k} $ be the system of simple roots in the relative root system $ \Phi _{k} (S,\ G) $ defined by the order relation chosen on $ X(S) $ ; it consists of the restrictions to $ S $ of the characters of the system $ \Delta $ . The Galois group $ \Gamma = \mathop{\rm Gal}\nolimits (k _{s} /k) $ acts naturally on $ \Delta $ , and the set $ \{ \Delta ,\ \Delta _{0} , \textrm{ the action of } \Gamma \textrm{ on } \Delta \} $ is called the $ k $ - index of the semi-simple group $ G $ . The role of the $ k $ - index is explained by the following theorem: Every semi-simple group over $ k $ is uniquely defined, up to a $ k $ - isomorphism, by its class relative to an isomorphism over $ k _{s} $ , its $ k $ - index and its anisotropic kernel. The relative root system $ \Phi _{k} (G) $ is completely defined by the system $ \Delta _{k} $ and by the set of natural numbers $ n _ \alpha $ , $ \alpha \in \Delta _{k} $ ( equal to 1 or 2), such that $ n _ \alpha \alpha \in \Phi _{k} (G) $ but $ (n _ \alpha + 1) \alpha \notin \Phi _{k} (G) $ . Conversely, $ \Delta _{k} $ and $ n _ \alpha $ , $ \alpha \in \Delta _{k} $ , can be determined from the $ k $ - index. In particular, two elements from $ \Delta \setminus \Delta _{0} $ have one and the same restriction to $ S $ if and only if they are located in the same orbit of $ \Gamma $ ; this defines a bijection between $ \Delta _{k} $ and the set of orbits of $ \Gamma $ into $ \Delta \setminus \Delta _{0} $ .


If $ \gamma \in \Delta _{k} $ , if $ O _ \gamma \subset \Delta \setminus \Delta _{0} $ is the corresponding orbit, if $ \Delta ( \gamma ) $ is any connected component in $ \Delta _{0} \cup O _ \gamma $ not all vertices of which lie in $ \Delta _{0} $ , then $ n _ \gamma $ is the sum of the coefficients of the roots $ \alpha \in \Delta ( \gamma ) \cap O _ \gamma $ in the decomposition of the highest root of the system $ \Delta ( \gamma ) $ in simple roots.

If $ k = \mathbf R $ , $ \overline{k} = \mathbf C $ , then the above relative root system and relative Weyl group are naturally identified with the root system and Weyl group, respectively, of the corresponding symmetric space.

References

[1] J. Tits, "Sur la classification des groupes algébriques semi-simples" C.R. Acad. Sci. Paris , 249 (1959) pp. 1438–1440 MR0106967
[2] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[3] J. Tits, "Classification of algebraic simple groups" , Algebraic Groups and Discontinuous Subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) pp. 33–62
How to Cite This Entry:
Relative root system. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Relative_root_system&oldid=44284
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article