Convexity properties of gradient maps

Abstract

We consider the action of a real reductive group G on a Kähler manifold Z which is the restriction of a holomorphic action of a complex reductive group H. We assume that the action of a maximal compact subgroup U of H is Hamiltonian and that G is compatible with a Cartan decomposition of H. We have an associated gradient map μ p : Z → p where g = k ⊕ p is the Cartan decomposition of g . For a G-stable subset Y of Z we consider convexity properties of the intersection of μ p ( Y ) with a closed Weyl chamber in a maximal abelian subspace a of p . Our main result is a Convexity Theorem for real semi-algebraic subsets Y of Z = P ( V ) where V is a unitary representation of U.