Connectedness of some fibres on a compact connected Lie group

Abstract

Given a compact connected Lie group G with Lie algebra , we prove the connectedness of the fibres for the following two maps:

1.	defined by g ↦ π(Ad(g)A), , where is a Cartan subalgebra of and is the orthogonal projection with respect to the G-invariant inner product ⟨·, ·⟩ on .
2.	ξ A,C : G → ℝ defined by g ↦ ⟨Ad (g)A, C⟩, .

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].

Keywords:

• connectedness
• C-numerical range
• compact connected Lie group

AMS Subject Classifications:

• Primary 22C05
• 15A60