On Hyperbolic Cones and Mixed Symmetric Spaces

A real semisimple Lie algebra g admits a Cartan involution, , for which the corresponding eigenspace decomposition g=k+p has the property that all operators ad X , Xp are diagonalizable over R .We call such elements hyperbolic, and the elements Xk are elliptic in the sense that ad X is semisimple with purely imaginary eigenvalues.The pairs (g,) are examples of symmetric Lie algebras, i.e., Lie algebras endowed with an involutive automorphism, such that the -1 -eigenspace of contains only hyperbolic elements.Let (g, ) be a symmetric Lie algebra and g= h+q the corresponding eigenspace decomposition for .The existence of "enough" hyperbolic elements in q is important for the structural analysis of symmetric Lie algebras in terms of root decompositions with respect to abelian subspaces of q consisting of hyperbolic elements.We study the convexity properties of the action of Inn g (h) on the space q .The key role will be played by those invariant convex subsets of q whose interior points are hyperbolic.