Page 128 - Textos de Matemática Vol. 44

118 X. LIU AND T.-Y. TAM
Example 3.10. Let g = sl3(R) and let C = diag(c,0,−c) with c > 0 and X = diag(1,0,−1). Then the image of fC,X : SO(3) → R is the interval [−2c, 2c] ⊂ R. By a result of Horn [15], C is the only element in the orbit O(C) (under the adjoint action of SO(3)) that is sent to 2c via the map Z → tr X Z . Thus f−1 (2c) is the centralizer of C in SO(3), i.e.,
C,X
f−1 (2c)={I3,diag(1,−1,−1),diag(−1,1,−1),diag(−1,−1,1)} C,X
which is evidently disconnected in SO(3). However if we consider sl3(C) with the same X and C, the corresponding f−1 (2c) is
C,X {diag(eiθ1,eiθ2,eiθ3):θ1 +θ2 +θ3 =0}⊂SU(3)
which is connected, as we already knew in the proof of Theorem 3.7.
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