Isotropy irreducible spaces and \(s\)-representations (Q1917374)

A homogeneous space \(G/K\) is called isotropy irreducible if the action of the isotropy group \(K\) on its tangent space is irreducible. Classical examples are the irreducible symmetric spaces. The non-symmetric isotropy irreducible spaces with \(K\) compact and connected have been classified by O. V. Manturov and J. A. Wolf. It turns out that \(G\) is compact and simple. If \(G= \text{SO}(n)\), \(\text{SU}(n)\) or \(\text{Sp}(n)\), these spaces are again related to symmetric spaces as had been observed by C. T. C. Wall at looking at the list of Wolf. The paper gives a new and simple conceptual proof for this relation. To this end a characterization of isotropy representations of symmetric spaces is given.