Connectedness of some fibres on a compact connected Lie group

Given a compact connected Lie group G with Lie algebra g, we prove the connectedness of the fibres for the following two maps: (1) π_A: G → t defined by g → π(Ad(g)A), A ε g, where t is a Cartan subalgebra of g and π: g → is the orthogonal projection with respect to the G-invariant inner product 〈·,·〉 on g. (2) ξ_A,C: G → ℝ defined by g → 〈Ad (g)A, C〉 A, Cεg. As a corollary of (2), we obtain a new proof of the convexity theorem in Tam [T.Y. Tam, Convexity of generalized numerical range associated with a compact Lie group, J. Austral. Math. Soc. 70 (2002), pp. 57-66].