Visible Actions on Reducible Multiplicity-Free Spaces

Abstract

The notion of (strongly) visible actions was introduced by T. Kobayashi [10, 11] for the biholomorphic action on a complex manifold with (possibly) infinitely many orbits. A holomorphic action of a Lie group G on a complex manifold D is strongly visible with a slice S if D = G ⋅ S, and there exists an anti-holomorphic and orbit preserving diffeomorphism σ on D such that σ|S = idS. Generalizing our previous result [22] about strongly visible actions on irreducible multiplicity-free spaces à la Kac, we treat reducible multiplicity-free spaces classified by Benson–Ratcliff and Leahy and prove that any multiplicity-free space naturally admits a strongly visible action. Further, we give an explicit description of S and σ. Our result gives an evidence to Kobayashi's conjecture [12, Conjecture 3.2] in the case of linear actions, asserting that we can take S to have the same dimension with the rank of V .