Classification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type

TL;DR

This paper classifies certain isoparametric submanifolds in non-compact symmetric spaces, showing they are principal orbits of Hermann type actions or isotropy actions under specific conditions, using building theory.

Contribution

It proves that isoparametric submanifolds with reflective focal submanifolds are principal orbits of Hermann type actions, extending the understanding of their geometric structure.

Findings

Isoparametric submanifolds with reflective focal submanifolds are Hermann type orbits.

Hyperpolar actions with reflective singular orbits are Hermann type actions.

Additional conditions identify submanifolds as isotropy orbits without real analyticity.

Abstract

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.