# Relative root system

*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://encyclopediaofmath.org/index.php?title=Relative_root_system&oldid=44284