Visible actions on the non-symmetric homogeneous space SO(8, ℂ)/G 2(ℂ)

Abstract

A holomorphic action of a Lie group G on a connected complex manifold D is called strongly visible with slice S if G · S is open in D and if there exists an antiholomorphic diffeomorphism σ of D preserving each G -orbit in D such that σ | s = id s . This paper deals with the non-symmetric spherical variety SO(8, ℂ)/ G 2 (ℂ). We prove that a maximal compact subgroup of SO(8, ℂ) acts on SO(8, ℂ)/ G 2 (ℂ) in a strongly visible fashion. Moreover, we can take a slice to be of dimension three which coincides with the rank of this spherical variety.