Page 126 - Textos de Matemática Vol. 44

116 X. LIU AND T.-Y. TAM
The above lemmas enable one to focus on the computation of the index

dim gα α∈Σ+,α(X)(w·α)(C)>0
of fC,X in (3.3) and the index α∈Σ+,α(X)(w·α)(C)<0
of −fC,X. If neither of them is 1, the convexity of WC(X,Y) follows for real semisimple Lie groups G with finite center (in which case K is compact). As an application, we have the convexity of the C-numerical range for complex semisimple Lie groups.
Theorem 3.7. Let G be a complex semisimple Lie group viewed as a real Lie group, and let g = k ⊕ p with p = ik be a Cartan decomposition of the (real) Lie algebra g of G. Then the C-numerical range WC (X, Y ) defined in (2.2) is convexforallC,X,Y ∈p.
Proof. Note that K is compact since k is a compact real form of g. Since g is
complex semisimple, each gα has even dimension over R [28, p. 217] and thus
the indices of fC,X and −fC,X are both even. By Atiyah’s lemma f−1 (c) is C,X
connected for all c ∈ R. Rotating WC (X, Y ) anti-clockwise by an angle θ ∈ R yields WC(X′,Y′), where
(X′, Y ′) = (cos θX + sin θY, − sin θX + cos θY ) ∈ p × p.
It follows that the intersection of WC(X,Y) with every straight line is con-
nected, whence WC (X, Y ) is convex.
Remark 3.8. Since p = ik in Theorem 3.7. It is essentially the same as the compact case discussed in Section 1. This gives a third proof of Tam’s result in [26].
The following example shows that the index condition is sufficient but not necessary for convexity of C-numerical range.
Example 3.9. Let g = sl3(R) and G = SL3(R). Let the Cartan involution θ bedefinedbyθ(X)=−X⊤ forallX∈g.Thenk=so(3),K=SO(3),andp is the space of all traceless symmetric matrices. Let a ⊂ p be the subspace of diagonal matrices. The root space decomposition of g relative to a is
g = a ⊕ REij . i̸=j
The root system is Σ = {ei − ej : 1 ≤ i ̸= j ≤ 3}. The centralizer M of a in K consists of diagonal matrices in SO(3), and the normalizer of a in K consists

dim gα