By Leila Figueredo

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C{^}. We have: anc c ^ 0 a for all a e $ (because * w o r k with the expression above in the 30 L. 3) = m (l-afpC)

C) E8 In this case: pCo^) = p(a3) = p(a2+a^) p(a2 ) = p(oL+a3+a,) p(a^) = p(a 5 ) = p(a 6 ) = p(a7 ) = p(ag ) = p C - ^ - ^ - ^ - a ^ - a . - a . - ^ - a g) and we obtain the following relations: pCo^) = p(a2)p(o^) 2 p(a2 ) = pCo^) p(a^) pCo^) p(a2)p(a^) = 1 from which we get that: pCo^) = p(a 2 ) = p(a^) = 1. ,8. (ii) Twisted Coxeter automorphisms (1) A , I odd Let s = -y- . I >_ 5 In this case the definition of p(oij) v implies that = p(a2 _ 1 ) = p(a2 ) = p(a2 _ 2 ) = ... -^) and we have the following relations: pCo^) = pCo^) p(a x ) " p(a g) s-1 P(a s> 2 = 1 and t h e r e l a t i o n s imply t h a t pCo^) = 1 p(as)2 = 1.

1). 3)). 5. * ~, x C Of £(V) . 4-a. h is central. So 0 *0 Clearly the element e e C*. 3)). Clearly the £(v)-module L(A) 1 k = —) (see Chapter 7). 4(2) we have that e = 1 and k = 1. So T c = c . 4) is a basic module for &(v) . 4, we have that Now what we have to show is that Let V for a e $, Z(a,£) = c Then ftw,N = Cv^, the L(X) 0' e €*. Let c f s, a e $ a ' vn on V. Hence, since is standard. We proceed indirectly. L(X) be a basic module for g_(v) . 1). 1 c1 = 1. 3), we see that be the highest weight vector of L(A).