304 INDEX Hilbert scheme, 104 Hochschild-Serre spectral sequence (HSSS), 88, 198 Hodge bundle, 9 Hodge bundles, 35 Hodge decomposition, 31 Hodge domain, 65 Hodge filtration, 31, 246 Hodge flag, 41 Hodge flags, 73 Hodge metric, 11 Hodge numbers, 34 Hodge representation, 51 Hodge structure of weight n, 31 Hodge structure of weight one, 8 Hodge tensors, 34 Hodge’s theorem, 32 Hodge-Riemann bilinear relations, 8, 33 holomorphic automorphic form of weight n, 17 holomorphic discrete series, 170 homogeneous complex manifold, 8 incidence variety I, 95, 149 infinitesimal character, 72 infinitesimal period relation (IPR), 44 infinitesimal variation of Hodge structure (IVHS), 46 isogeny, 15 Iwasawa decomposition, 130 K-type, 83, 85, 87 Kostant class, 92, 225 Kostant form, 162 L-graded decomposition, 227 L2-cohomology group, 83 Lagangian Grassmannian GrL(h, Ck), 35 Lagrange flag, 43 Lagrange line, 43, 44 Lagrange quadrilateral, 151 length l(w) of an element w ∈ W , 228 Levi form (intrinsic), 142 Levi form (of an exhaustion function), 96 limiting mixed Hodge structure (LMHS), 248 limits of discrete series (LDS), 69 limits of discrete series for SL2(R), 14 local cohomology, 117 log analytic variety with slits, 273 logarithmic growth, 18 Matsuki duality, 14, 25, 111 Matsuki orbit triple, 124 Maurer-Cartan equation, 75, 174 Maurer-Cartan matrix, 76 mirror quintic variety, 42 mixed Hodge structure, 245 mixed Hodge structure split over R, 246 modular form, 18 monodromy group, 45 moving frame, 173 Mumford-Tate group, 36 Mumford-Tate group of a variation of Hodge structure, 45 neat (subgroups), 69 negative line bundle, 12 nilpotent cone, 222 nilpotent orbit, 249 nilpotent orbit theorem, 270 non-classical automorphic cohomology group, 165 non-classical flag domain, 27, 77 non-compact roots, 62 parabolic subgroup, 35 parabolic Picard modular forms, 236 Penrose transform, 23, 27, 162, 173, 181 period domain, 34 period domain Hodge structures of weight one, 8 period matrix, 6, 35 periods, 6 Picard modular forms, 173, 293 Plancherel formula, 69 polarized Hodge structure, 33 polarized Hodge structure of weight one, 8 positive line bundle, 12 positive root, 19 primitive decomposition, 248 primitive space, 248 projective frame, 149 property W, 194 pseudo-concavity, 241 q(μ), 77 qc(μ + ρ), 91 quartets, 122 rank of G, 52 rationally connected complex manifold, 237 real form (of a complex Lie group, 14 reductive subgroup, 75 reference flag, 167 regular weight, 23, 69 relative differential, 153 restricted root system, 105, 129 restricted roots, 121 restriction of scalars, 54 root lattice, 19, 53 root spaces, 55 root vector, 19 root vectors, 55 (σ, θ)-stable, 121 σ-nilpotent orbit, 272 Schubert cell, 228

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