Page 120 - Textos de Matemática Vol. 44

110 X. LIU AND T.-Y. TAM
2. Numerical range and semisimple Lie algebra
2.1. Star-shapedness and complex semisimple Lie algebra. Tam [25] considered a generalized C-numerical range in the context of complex semisim- ple Lie algebras. Let g be a complex semisimple Lie algebra and let k be a compact real form of g. Then g = k ⊕ ik is a Cartan decomposition of g with a corresponding Cartan involution θ. The Killing form B induces an inner product Bθ on g defined by
Bθ(X,Y)=−B(X,θY), ∀X,Y ∈g.
Let G be a connected complex Lie group with Lie algebra g and let K be the analytic subgroup of G with Lie algebra k. Given X,C ∈ g, the C-numerical range of X is defined by
(2.1) WC(X) = {Bθ(C,Ad(k)X) : k ∈ K}.
Note that the usual C-numerical range is for the reductive Lie algebra gln(C) and that the compact case is essentially a special one with X ∈ k. Tam [25] conjectured that for any X ∈ g and f ∈ g∗, the dual space of g, the set f(Ad(K)X) is star-shaped with respect to the origin.
The adjoint orbit Ad (K)X depends only on Ad GK, the analytic subgroup of the adjoint group Intg corresponding to adgk, and thus Ad(K)X is inde- pendent of the choice of G. Let t be a maximal abelian subalgebra of k. Then h = t⊕it is a Cartan subalgebra of g. Let g = h⊕ gα be the root space
α∈∆ decomposition [16, p.35] of g with respect to h, where
∆ := {α∈h∗ :α̸=0and dimgα ̸=0},
gα := {X∈g:[H,X]=α(H)XforallH∈h}.
Since B(gα,gβ) = 0 whenever α+β ̸= 0, we have the orthogonal projection π : g → h. Cheung and Tam [6] proved that π(Ad(K)X) is star-shaped in h with star center 0 for all X ∈ g. They further affirmed Tam’s conjecture for the complex simple Lie algebras of type B [6]. The conjecture is valid for simple Lie algebras of type A [7], D, E6 and E7 [9]; it remains unknown for type C, E8, F4 and G2.
2.2. Convexity and real semisimple Lie algebra. Li and Tam [19] gen- eralized C-numerical range in the context of real semisimple Lie algebras. Let g be a real semisimple Lie algebra. Let G be a connected real semisimple Lie group with Lie algebra g. Let g = k ⊕ p be the Cartan decomposition of g corresponding to a Cartan involution θ, where k and p are the +1 and −1 eigenspaces of θ, respectively. The Killing form B is positive definite on p and