Weyl group

The Weyl group of symmetries of a root system. Depending on the actual realization of the root system, different Weyl groups are considered: Weyl groups of a semi-simple splittable Lie algebra, of a symmetric space, of an algebraic group, etc.

Let $ G $ be a connected affine algebraic group defined over an algebraically closed field $ k $ . The Weyl group of $ G $ with respect to a torus $ T \subset G $ is the quotient group $$ W(T,\ G) = N _{G} (T) / Z _{G} (T), $$ considered as a group of automorphisms of $ T $ induced by the conjugations of $ T $ by elements of $ N _{G} (T) $ . Here $ N _{G} (T) $ is the normalizer (cf. Normalizer of a subset) and $ Z _{G} (T) $ is the centralizer of $ T $ in $ G $ . The group $ W(T,\ G) $ is finite. If $ T _{0} $ is a maximal torus, $ W( T _{0} ,\ G) $ is said to be the Weyl group $ W(G) $ of the algebraic group $ G $ . This definition does not depend on the choice of a maximal torus $ T _{0} $ ( up to isomorphism). The action by conjugation of $ N _{G} ( T _{0} ) $ on the set $ B ^ {T _{0}} $ of Borel subgroups (cf. Borel subgroup) in $ G $ containing $ T _{0} $ induces a simply transitive action of $ W( T _{0} ,\ G) $ on $ B ^ {T _{0}} $ . The action by conjugation of $ T $ on $ G $ induces an adjoint action of $ T $ on the Lie algebra $ \mathfrak g $ of $ G $ . Let $ \Phi (T,\ G) $ be the set of non-zero weights of the weight decomposition of $ \mathfrak g $ with respect to this action, which means that $ \Phi (T,\ G) $ is the root system of $ \mathfrak g $ with respect to $ T $ ( cf. Weight of a representation of a Lie algebra). $ \Phi (T,\ G) $ is a subset of the group $ X(T) $ of rational characters of the torus $ T $ , and $ \Phi (T,\ G) $ is invariant with respect to the action of $ W(T,\ G) $ on $ X(T) $ .

Let $ G $ be a reductive group, let $ Z(G) ^{0} $ be the connected component of the identity of its centre and let $ T _{0} $ be a maximal torus of $ G $ . The vector space $$ X(T _{0} /Z(G) ^{0} ) _ {\mathbf Q} = X(T _{0} /Z(G) ^{0} ) \otimes _ {\mathbf Z} \mathbf Q $$ is canonically identified with a subspace of the vector space $$ X(T _{0} ) _ {\mathbf Q} = X(T _{0} ) \otimes _ {\mathbf Z} \mathbf Q . $$ As a subset of $ X {( T _{0} )} _ {\mathbf Q} $ , the set $ \Phi ( T _{0} ,\ G) $ is a reduced root system in $ X( T _{0} /Z(G) ^{0} ) _ {\mathbf Q} $ , and the natural action of $ W( T _{0} ,\ G) $ on $ {X( T _{0} )} _ {\mathbf Q} $ defines an isomorphism between $ W( T _{0} ,\ G) $ and the Weyl group of the root system $ \Phi (T _{0} ,\ G) $ . Thus, $ W(T _{0} ,\ G) $ displays all the properties of a Weyl group of a reduced root system; e.g. it is generated by reflections (cf. Reflection).

The Weyl group of a Tits system is a generalization of this situation (for its exact definition see Tits system).

The Weyl group $ W $ of a finite-dimensional reductive Lie algebra $ \mathfrak g $ over an algebraically closed field of characteristic zero is defined as the Weyl group of its adjoint group. The adjoint action of $ W $ in the Cartan subalgebra $ \mathfrak p $ of $ \mathfrak g $ is a faithful representation of $ W $ . The group $ W $ is often identified with the image of this representation, being regarded as the corresponding linear group in $ \mathfrak p $ generated by the reflections. The concept of a "Weyl group" was first used by H. Weyl

in the special case of finite-dimensional semi-simple Lie algebras over the field of complex numbers. A Weyl group may also be defined for an arbitrary splittable semi-simple finite-dimensional Lie algebra, as the Weyl group of its root system. A relative Weyl group may be defined for an affine algebraic group $ G $ defined over an algebraically non-closed field. If $ T $ is a maximal $ k $ - split torus of $ G $ , then the quotient group $ N _{G} (T)/ Z _{G} (T) $ ( the normalizer of $ T $ over its centralizer in $ G $ ), regarded as the group of automorphisms of $ T $ induced by the conjugations of $ T $ by elements of $ N _{G} (T) $ , is said to be the relative Weyl group of $ G $ .

For the Weyl group of a symmetric space, see Symmetric space. The Weyl group of a real connected non-compact semi-simple algebraic group is identical with the Weyl group of the corresponding symmetric space. For the affine Weyl group see Root system.

References

[1a]	H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen I" Math. Z. , 23 (1925) pp. 271–309 MR1544744
[1b]	H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen II" Math. Z. , 24 (1925) pp. 328–395 MR1544744
[2]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[3]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[4]	N. Bourbaki, "Lie groups and Lie algebras" , Elements of mathematics , Hermann (1975) (Translated from French) MR2109105 MR1890629 MR1728312 MR0979493 MR0682756 MR0524568 Zbl 0319.17002
[5a]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. I.H.E.S. , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[5b]	A. Borel, J. Tits, "Complément à l'article "Groupes réductifs" " Publ. Math. I.H.E.S. , 41 (1972) pp. 253–276 MR0315007
[6]	F. Bruhat, J. Tits, "Groupes algébriques simples sur un corps local" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 23–36 MR0230838 Zbl 0263.14016
[7]	S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101

Comments

The affine Weyl group is the Weyl group of an affine Kac–Moody algebra. One may define a Weyl group for an arbitrary Kac–Moody algebra.

The Weyl group as an abstract group is a Coxeter group.

Weyl groups play an important role in representation theory (see Character formula).

References

[a1]	J. Tits, "Reductive groups over local fields" A. Borel (ed.) W. Casselman (ed.) , Automorphic forms, representations and -functions , Proc. Symp. Pure Math. , 33:1 , Amer. Math. Soc. (1979) pp. 29–69 MR0546588 Zbl 0415.20035
[a2]	J.E. Humphreys, "Reflection groups and Coxeter groups" , Cambridge Univ. Press (1991) MR1066460 Zbl 0768.20016 Zbl 0725.20028

The Weyl group of a connected compact Lie group $ G $ is the quotient group $ W = N/T $ , where $ N $ is the normalizer in $ G $ of a maximal torus $ T $ of $ G $ . This Weyl group is isomorphic to a finite group of linear transformations of the Lie algebra $ \mathfrak t $ of $ T $ ( the isomorphism is realized by the adjoint representation of $ N $ in $ \mathfrak t $ ), and may be characterized with the aid of the root system $ \Delta $ of the Lie algebra $ \mathfrak g $ of $ G $ ( with respect to $ \mathfrak t $ ), as follows: If $ \alpha _{1} \dots \alpha _{r} $ is a system of simple roots of the algebra, which are linear forms on the real vector space $ \mathfrak t $ , the Weyl group is generated by the reflections in the hyperplanes $ \alpha _{i} (x) = 0 $ . Thus, $ W $ is the Weyl group of the system $ \Delta $ ( as a linear group in $ \mathfrak t $ ). $ W $ has a simple transitive action on the set of all chambers (cf. Chamber) of $ \Delta $ ( which, in this case, are referred to as Weyl chambers). It should be noted that, in general, $ N $ is not the semi-direct product of $ W $ and $ T $ ; all the cases in which it is have been studied. The Weyl group of $ G $ is isomorphic to the Weyl group of the corresponding complex semi-simple algebraic group $ G _{\mathbf C} $ ( cf. Complexification of a Lie group).

A.S. Fedenko