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