Visible Actions on Irreducible Multiplicity-Free Spaces

A holomorphic action of a Lie group G on a connected complex manifold D is called strongly visible with a slice S if D′ ≔ G · S is open in D and there exists an antiholomorphic and orbit-preserving diffeomorphism σ of D′ such that σ |S =  id S. In this article, we study linear, strongly visible actions. We prove that irreducible multiplicity-free space V of a connected compact Lie group is strongly visible. Furthermore, we find an explicit description of S and σ according to Kac's classification. Our result gives an evidence to Kobayashi's conjecture [10, Conjecture 3.2] in the case of irreducible multiplicity-free spaces, asserting that we can take S to have the same dimension as the rank of V.