Index a representation that leads to a Hodge representation, 53 anisotropic maximal torus, 38 anti-dominant Weyl chamber, 71 arithmetic structure (on automorphic cohomology, 221 automorphic cohomology, 154 basic diagram, 149 Beilinson-Bernstein localization, 117 Blattner parameter, 202 Borel subgroup, 19 Borel-Weil-Bott theorem, 23 Bruhat cell, 181 canonical extension of the Hodge bundle, 18 Cartan decomposition, 53 Cartan involution, 53 Cartan sub-algebra, 19, 55 Cartan subgroup, 35 Cartan-Killing form, 55 Casimir operator, 200 Casselman-Osborne lemma, 194 Cauchy-Reimann subspace (of the tangent space), 135 Cauchy-Riemann tangent space, 99, 133 Cayley transform, 124, 254 center of the universal enveloping algebra, 70 character, 52 Chern form, 11, 75 Chow’s theorem, 238 classical flag domain, 27, 77 CM field, 48 co-character, 52 co-root vectors, 55 compact dual ˇ of a period domain D, 34 compact dual of the period domain for Hodge structures of weight one, 8 compact roots, 62 completion of a variation of Hodge structure, 222 complex multiplication (CM) Hodge structure, 37, 46 congruence subgroup, 16 conservation law (for orbit codimensions, 128 correspondence diagram, 22 correspondence space, 21 correspondence space W, 95, 148 correspondence spaces, 27 cusp, 17 cusp form, 18 cuspidal automorphic cohomology, 221 cycle space U, 95, 102 cycle spaces, 27 Deligne splitting (of a Hodge structure), 246 differential form, type of, 7 discrete series (DS), 69 discrete series for SL2(R), 14 dominant Weyl chamber, 55 dual pair, 111 enhanced flag variety, 147 exhaustion function , 96 exhaustion function modulo G R , 108 expanding cohomology about Z, 86 fan, 222, 272 flag domain, 14, 72 flag variety, 14, 72 (gC, KC)-module, 71 grading, 53 grading associated to a filtration, 246 grading element, 227 Grauert domain, 107 Grothendieck residue symbol, 120 Gysin map, 33 Harish-Chandra module, 14, 70 Harish-Chandra parameter, 83, 202 Hermitian symmetric domain, 77 Hermitian type (of G), 109 Hermitian vector bundle, 35 highest weight, 56 highest weight vector, 72 303
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