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abelian assume automorphism Bergman kernel Borel Borel subgroup boundary bounded bundle character codimension cohomology cokernel compact compute convex convex hull Corollary corresponding curve cyclic defined definition deleted foliation denote diffeomorphism discrete series representation divisor domain element equation exact sequence exists extreme polygon finite fixed follows formula function given Hamiltonian Hence Holder continuously homology homomorphism homotopy equivalence Hopf algebra implies induced inequality integer invariant irreducible isomorphism Lemma linear Lipschitz loop Lorentz metric manifold mapping cylinder Math metric minimal surfaces module nilpotent notation obtain operator oriented Plateau's problem prime projection proof of Theorem Proposition prove regular neighborhood representation resp restriction result roots satisfies Schubert variety Section solution space subgroup subset summand Suppose tangent cones trivial unique vanishes vector vertex vertices Weyl group zero