INVARIANT CONVEX SETS IN POLAR REPRESENTATIONS

Abstract. We study a compact invariant convex set E in a polar rep-resentation of a compact Lie group. Polar rapresentations are given by the adjoint action of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g = k ⊕ p. If a ⊂ p is a maximal abelian subalgebra, then P = E ∩ a is a convex set in a. We prove that up to conjugacy the face structure of E is completely determined by that of P and that a face of E is exposed if and only if the corresponding face of P is exposed. We apply these results to the convex hull of the image of a restricted momentum map. The boundary of a compact convex set is the union of its faces. Among the faces, the simplest ones are the exposed ones. They are given by the intersection of the convex set with a supporting hyperplane. In [3, 4] we studied the convex hull O ̂ of a K-orbit O in p, where p is given by the Cartan decomposition g = k ⊕ p of a reductive Lie algebra g and K acts on p by the adjoint representation. In this paper we use the results of [4] and show that a substantial part of them holds for any K–invariant compact convex set E of p. More precisely we study the faces of E. We show in Proposition 1.2 that for a face F of E there exists a subalgebra s ⊂ p such that F is a subset of ps = {x ∈ p: [x, s] = 0} and F is invariant with respect to the action of Ks = {h ∈ K: Ad(h)(s) = s}, where Ad denotes the adjoint representation. If we fix a maximal abelian subalgebra a ⊂ p, then the set P = E ∩ a is convex and invariant with respect to the action of the normalizer NK(a) = {h ∈ K: Ad(h)(a) = a} of a in K. The NK(a)–action on P induces an action on the set of faces of P. Similarly K acts on the set of faces of E. Denote these sets by F (P) respectively by F (E). If σ is a face of P, let σ⊥ denote the orthogonal complement in a of the affine hull of σ (see Section 1). Our main result is Theorem 0.1. The map F (P) → F (E), σ 7 → Kσ ⊥ · σ is well–defined and induces a bijection between F (P)/NK(a) and F (E)/K