Invariant convex sets in polar representations (Q314410)

The structure of faces of convex bodies, which are invariant under some actions of compact Lie groups, is investigated. The notion of exposed faces of convex bodies is used. Not all faces are exposed one's. Let \(g\) be a semisimple real Lie algebra, \(k\) be its maximal compact subalgebra and \(g=k+p\) be the Cartan decomposition. The Lie group \(K\), corresponding to \(k\), acts on \(p\) by the adjoint representation. Let \(E\) be some \(K\)-invariant compact body in \(p\). For maximal abelian subalgebra \(a \subset p\) the intersection \(P = E \cap a\) is considered; it is convex. Let \(N_K(a)\) is the normalizer of \(a\) in \(K\). The main result of this paper is the follows. Let \(\mathcal F (E)\) is the set of all faces of the convex body \(E\). Then there is a natural map \(\mathcal F(P) \to \mathcal F(E)\) which induces the bijection \(\mathcal F(P)/N_K(a) \to \mathcal F(E)/K\). So up to conjugacy the face structure of \(E\) is completely determined by that of \(P\). As an application it is proved that the face of \(E\) is exposed if and only if the corresponding face of \(P\) is exposed. Also as an application the convex hull of the image of a restricted momentum map (or gradient map image) is investigated.