# Abelian Group Theory by D. Arnold, R. Hunter, E. Walker

By D. Arnold, R. Hunter, E. Walker

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12), l are sets of the following types: type a. An orbit of p6 i, where 6 i is a f u n d a m e n t a l weight, 6 i ~ N. type b. The union of (0) and the orbit of p~i, where ~i is a fund a m e n t a l weight, ~i 6 N. Proof of (1). First we note that for every ~ C N, y ~ &, the weights y and way = y -

The r e p r e s e n t a t i o n (Pi)M1. 0 Fr i n LM . z d e n o t e s l the vector space LM. , viewed as r e p r e s e n t a t i o n space of oi). 9). denote (see Lemma 5O So every non-zero weight of Pi is in the orbit of ~i, and has multiplicity 1 (see [ 7 [, Expos@ 16, Proposition 20, Proposition 1 and Expos@ 4). Suppose zero is a weight of Pi" Then 6 i is Z-connected W6 i U (0), so 6i is a multiple a root, because of a root. it is a minimal formula, or from [25]. In fact ~i has to be dominant weight.

Z*~,6 = such that ~+8 is on ~+8. It is c l e a r : 0). Hence while ~+8 we that consider = Y+~ is n. 6, suppose (iii)). = < ~ + 8 , 8 > = 2. So