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a-root algebra apply assume basic character boundary Cartan Cartan subalgebra Cayley transform closed geodesic compact component condition constant contains convex Corollary CR structure curvature defined denote diffeomorphism dimension discrete series discrete series representation divisor embedded exact sequence exists extremal ray fiber filtration finite follows formula function given Hence homomorphism homotopy hyperbolic implies induced from discrete infinitesimal character integral irreducible isomorphism Jordan blocks kernel Lemma limit of discrete linear manifold Math minimal surfaces module morphism multiple neighborhood non-singular noncompact nonzero norm notation numerically effective o o o o o o obtain operator orthogonal parabolic subgroup positive projective projective module proof of Theorem properties Proposition prove real analytic representation resp result Ricci curvature root satisfies Schmid identity Section subgroup subset subspace Suppose tempered representations Theorem 1.1 vector space Weyl Weyl group XX XX zero