Page 127 - Textos de Matemática Vol. 44

CONNECTEDNESS, HESSIANS AND GENERALIZED NUMERICAL RANGES 117
of generalized permutation matrices in SO(3) whose nonzero entries are either 1 or −1. Thus the Weyl group W = M′/M is isomorphic to the group P3 of permutation matrices. Let a+ ⊂ a be the fundamental Weyl chamber consisting of all diagonal matrices whose diagonal entries are in descending order. The set of positive roots is then Σ+ = {e1 −e2, e2 −e3, e1 −e3}. Now pick C, X ∈ a+ and consider the map fC,X : SO(3) → R as defined in (3.1). Obviously KC = KX = M. By Lemma 3.2, the critical manifold of fC,X is KC,X = KCWKX = M′. Because α(X) > 0 and dimgα = 1 for all α ∈ Σ+, the index given by (3.3) is equal to the number of positive roots sent to positive roots by the w ∈ W under consideration. Since WC and WX are trivial, each Weyl group element can appear for some k ∈ KC,X . Therefore we have the following six cases for the indexoffC,X withthenotationsthatα1 :=e1−e2,α2 :=e2−e3,α3 :=e1−e3.
1 0 0
Case1:w=0 1 0.Sincew·α1 =α1,w·α2 =α2,w·α3 =α3,the
001
index is 3.
0 1 0
Case2:w=1 0 0.Sincew·α1 =−α1,w·α2 =α3,w·α3 =α2,the
001
index is 2.
1 0 0
Case3:w=0 0 1.Sincew·α1 =α3,w·α2 =−α2,w·α3 =α1,the
010
index is 2.
0 0 1
Case4:w=0 1 0.Sincew·α1 =−α2,w·α2 =−α1,w·α3 =−α3,
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the index is 0.
0 0 1
Case5:w=1 0 0.Sincew·α1 =α2,w·α2 =−α3,w·α3 =−α1,
010
the index is 1.
0 1 0
Case6:w=0 0 1.Sincew·α1 =−α3,w·α2 =α1,w·α3 =−α2,
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Now dim K = dim SO(3) = 3. This shows that for g = sl3 (R) the condition
the index is 1.
of Lemma 3.6 is not satisfied, but WC (X, Y ) is still convex [5].
The following example shows that WC(X,Y) is convex but some fibre of fC,X is not connected for some real semisimple Lie algebra.