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