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KOSTANT CONVEXITY FOR SYMMETRIC R-SPACES REVISITED

An embedding of a connected and compact manifold M into Euclidean space (V, 〈 , 〉) is called extrinsically symmetric if for any x ∈ M , the reflection rx of V about the affine normal space at x leaves M invariant. If we equip M with the submanifold metric, then the restriction of rx to M is an involutive isometry. Relative to these isometries, M is an (intrinsically) symmetric space. One knows precisely what these embeddings are: by a theorem of Ferus [3] (see also [4]), they are the so-called symmetric R-spaces, which we present in what follows. We start with a simply connected symmetric space of non-compact type G/K and consider the Lie algebras of G and K, which are g and k, respectively. Let also g = k ⊕ p be the corresponding Cartan decomposition and a ⊂ p a maximal abelian subspace. The roots are elements of a∗; we denote by W be the corresponding Weyl group, which acts linearly on a. Pick ξ ∈ a such that α(ξ) ∈ {−1, 0, 1}, for any root α. The adjoint orbit Kξ := AdG(K)ξ turns out to be an extrinsic symmetric submanifold of p, the latter being equipped with the inner product given by the negative of the Killing form. Such orbits are called symmetric R-spaces. The aforementioned theorem of Ferus says that all extrinsic symmetric submanifolds in Euclidean space are of this type.