1. INTRODUCTION S/Fs = T/Sf. Clearly S = 5/(5(1 A) and S = s/7r(A), where n: t 5 is orthogonal projection. 1.1. Preliminaries Let $ be a root system on a real vector space V. Fix a set of simple roots A for $. Denote by Q = Q($) C F* the root lattice, P = P($) C V* the lattice of weights with basis {ma}ae&. Further, we denote the inverse root system by $ v C V. We have Qv = Q($v) c V the coroot lattice. It is dual to P and the coweight lattice P v C V is dual to Q($). We denote the Weyl group by W($) or simply by VF if ^ is clear from the context. We fix an inner product (•, •) on V invariant under the Weyl group and use it to identify V with V*. We choose this inner product so that in each irreducible factor the shortest length of a coroot is \/2. Coroots of this length are called short coroots whereas coroots of longer lengths are called long coroots. Roots dual to short coroots are thus long roots and roots dual to long coroots are short roots. Associated to a, b G $ is the Cartan integer defined by (1.1) n ( a , 6 ) = _ _ The Cartan integers for pairs of elements in A determine and are determined by the Dynkin diagram D(f$) together with the multiplicities and arrows of its bonds [4]. The set A v C $ v consists of the coroots a v inverse to each root a G A. The Cartan integers n(av,bv) for a v ,6 v G A v are described by a Dynkin diagram Dv(£), the coroot diagram for £. Its nodes are identified in the obvious way with the nodes of D(£). Its bonds, including the multiplicities, are exactly the same as the bonds in .D($), but the direction of every arrow is reversed. Suppose that $ is irreducible. Let d be the highest root of $ with respect to A. Set a = —d and let A = A U {a} be the extended set of simple roots. Let Co C V be the positive Weyl chamber associated to A and let A C Co be the unique alcove in Co containing the origin. The walls of A are given by {a = 0}aeA and {a = —1}. Hence there is a natural bijection between A and the walls of A. The set A is the set of nodes for the extended Dynkin diagram D($). The Cartan integers n(a, b) for a, b G A are recorded in the multiplicities of the bonds and the directions of the arrows of £)(£), by exactly the same rules as given in the case of D(&). (In the case of Ai, we shall always make the convention that the two nodes are connected by two single bonds, so that the diagram is a cycle.) Dually, there is the extended coroot diagram Dv(&) whose nodes are the coroots A v inverse to the roots in A. (N.B. The coroot inverse to the highest root is the highest short coroot, and is equal to the highest coroot if and only if $ is simply laced.) As in the case of -D(1), the diagram Dv(&) is obtained from D($) by reversing the directions of all the arrows on the multiple bonds. In case $ = ]J^ &{ is reducible and the £j are the irreducible factors, we define the set of extended roots A as ]Ji A , and define the extended root and coroot diagrams as the disjoint union of the corresponding diagrams of the factors.
Previous Page Next Page