A characterization of non-tube type Hermitian symmetric spaces by visible actions

Abstract

Let G/K be an irreducible Hermitian symmetric space of non-compact type, and \({G_{\mathbb{C}}/K_{\mathbb{C}}}\) its complexification by forgetting the original complex structure. Then, \({D :=G_{\mathbb{C}}/[K_{\mathbb{C}}, K_{\mathbb{C}}]}\) is a non-symmetric Stein manifold. We prove that a maximal compact subgroup of \({G_{\mathbb{C}}}\) acts on D in a strongly visible fashion in the sense of Kobayashi (Publ Res Inst Math Sci 41:497–549, 2005) if and only if G/K is of non-tube type. Our proof uses the theory of multiplicity-free representations and a construction of a slice and an anti-holomorphic involution on D.