Index abelian category, 116, 117 abelian subcategory, 118 abelian variety, 421 acyclic objects for a functor, 130 sheaves, 159 additive category, 116 additive functor, 116 adjacent Weyl chambers, 479 adjoint of an operator, 360 adjoint representation, 384 adjunction morphisms, 155 admissible representation, 485 afRne variety, 186 algebraic element, 62 field extension, 62 group, 376, 419 group action, 424 prevariety, 186 sheaf, 194 subvariety, 45 variety, 187 algebraically induced analytic sheaves, 347 aligned pair of boxes, 249 ample sheaf, 326 analytic modules of finite type, 487 analytic polyhedron, 261 analytic sheaf, 216 analytic vector, 486 Arens-Royden theorem, 296 Artin-Rees theorem, 197 balanced set, 264, 304 Banach algebra, 245, 288 bidegree of a differential form, 238 biholomorphic equivalence, 51 mapping, 50, 55 biholomorphically equivalent, 56 biregular mapping, 55 Borel subalgebra, 399, 443 Borel subgroup, 443 Borel-Weil theorem, 477 Borel-Weil-Bott theorem, 480 bounded complex, 119 subset ofH(U), 30 bounded subset, 264 b o x i n C n , 249 branching order, 73 Cartan subalgebra, 395 Cartan's factorization lemma, 251 Cartan's first criterion, 391 Cartan's second criterion, 391 Cartan's Theorem A, 278 Cartan's Theorem B, 279 Cartan-Serre theorem, 302 Cartesian product of prevarieties, 187 Casimir operator, 469 category, 114 abelian, 117 additive, 116 homotopy , 120 of functors, 115 of modules, 133 of morphisms, 115 Cauchy integral formula generalized, 8

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