Classification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type
In this paper, we assume that all isoparametric submanifolds have flat section. The main purpose of this paper is to prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type admits a reflective focal submanifold and if it of real analytic, then it is a principal orbit of a Hermann type action on the symmetric space. A hyperpolar action on a symmetric space of non-compact type admits a reflective singular orbit if and only if it is a Hermann type action. Hence is not extra the assumption that the isoparametric submanifold admits a reflective focal submanifold. Also, we prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type satisfies some additional conditions, then it is a principal orbit of the isotropy action of the symmetric space, where we need not impose that the submanifold is of real analytic. We use the building theory in the proof.