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2) (ii)) we have that c -a is determined by c . 2) is determined by its values on a base p: by L •> C * A of $. So we define 32 L. FIGUEIREDO P( and the condition that p(a) = mc a p(va) = p(a) for all p be a group homomorphism from a e $. for all Conversely, let p(va) = p(a) Since c a = c va for all L to C*. Then a e •$ we have that a e L. ,£ i c fs a e L. 2) (ii) and (iii). c a = —p(a), it is easy m // 8. COXETER AND TWISTED COXETER AUTOMORPHISMS This section is independent of 2-7. 8) stated in section 2.

Lepowsky, S. Mandelstam and I. M. Singer, Publ. Mathematical Sciences Research Institute #3, Springer-Verlag (1985), 231-273. [14] D. Mitzman, Integral bases of affine Lie algebras and their universal enveloping algebras. American Mathematical Society, Contemporary Mathematics vol. 40 (1985). 57 58 L. FIGUEIREDO [15] G. Segal, Unitary representations of some infinite-dimensional groups. Comm. Math. Phys. 8£ (1981), 301-342. [16] T. A. Springer, Regular elements of finite reflection groups. Math.

P,(va) = p,(a) A A p, : L -> (C* for all a e L and i be the root lattice of a root system $ of type A, D be a Coxeter automorphism or a twisted Coxeter automor- The number of distinct group homomorphisms p: L -> C* such that p(va) = p(cx) is equal to the number of inequivalent basic modules for &(v) . ,(Xp} of $. lfs among the integers (i) Coxeter automorphisms. (see Table III). p(va) = p(a) for all a e L we must have: p(a ) = ... -aj. ). Thus there are exactly (2) Hence p(ax) p(a-) = e, where e Z+l is a = 1.