Introduction ................................................... xi
Part I Introduction to Flag Domain Theory
1 Structure of Complex Flag Manifolds ......................... 5
1.1 Structure theory and the root decomposition ............. 5
1.2 Cartan highest weight theory ............................ 9
1.3 Borel subgroups and subalgebras ........................ 14
1.4 Parabolic subgroups and subalgebras .................... 17
1.5 Homogeneous holomorphic vector bundles ................. 20
1.6 The Bott-Borel-Weil Theorem ............................ 22
2 Real Group Orbits .......................................... 27
2.1 Bruhat. Lemma and an application ....................... 27
2.2 Real isotropy .......................................... 28
3 Orbit Structure for Hermitian Symmetric Spaces ............. 31
3.1 Strongly orthogonal roots .............................. 31
3.2 Orbits and Cayley transforms ........................... 33
4 Open Orbits ................................................ 37
4.1 Automorphisms and regular elements ..................... 37
4.2 Fundamental Cartan subalgebras and open G0-orbits ...... 38
4.3 Compact subvarieties of open orbits .................... 40
4.4 Holomorphic functions .................................. 42
4.5 Measurabilily of open orbits ........................... 44
4.6 Background on Levi geometry ............................ 46
4.7 The exhaustion function for measurable open orbits ..... 50
5 The Cycle Space of a Flag Domain ........................... 55
5.1 Definitions and first properties ....................... 55
5.2 The three cases ........................................ 57
5.3 Cycle spaces of measurable open orbits are Stein ....... 58
5.4 The cycle space in the hermitian case .................. 60
5.5 The classical hermitian case ........................... 66
Part II Cycle Spaces as Universal Domains
6 Universal Domains .......................................... 77
6.1 Definitions and first properties ....................... 78
6.2 Adapted structure ...................................... 83
6.3 Invariant CR structure and pseudoconvexity ............. 84
7 B-Invariant Hypersurfaces in Mz ............................ 93
7.1 Iwasawa-Borel subgroups and their Schubert varieties ... 94
7.2 Envelope construction .................................. 96
7.3 Schubert intersection properties ...................... 101
7.4 Trace transform ....................................... 104
8 Orbit Duality via Momentum Geometry ....................... 113
8.1 Coadjoint orbits ...................................... 114
8.2 The K0-energy function ................................ 116
8.3 Duality ............................................... 121
8.4 Orbit ordering ........................................ 123
9 Schubert Slices in the Context of Duality ................. 125
9.1 Schubert slices in arbitrary G0-orbits ................ 126
9.2 Supporting hypersurfaces at the boundary of MD ........ 130
10 Analysis of the Boundary of U ............................. 133
10.1 Preparation .......................................... 135
10.2 Linearization and the Jordan decomposition ........... 141
10.3 Characterization of closed orbits .................... 145
10.4 The slice theorem and related isotropy
computations ......................................... 148
10.5 Example: Two-dimensional affine quadric .............. 152
10.6 sl2 models at generic points of bd(U) ................ 154
11 Invariant Kobayashi-Hyperbolic Stein Domains .............. 163
11.1 Hyperbolicily of domains in G/K ...................... 164
11.2 The maximal invariant Kobayashi-hyperbolic Stein
domain in an sl2-model ............................... 169
11.3 Maximality and the characterization of cycle
domains .............................................. 171
12 Cycle Spaces of Lower-Dimensional Orbits .................. 175
12.1 Definition of cycle space ............................ 176
12.2 Intersection with Schubert varieties ................. 177
12.3 Hypersurfaces in the complement of the cycle space ... 180
12.4 Cycle Spaces of nonclosed orbits ..................... 181
12.5 Cycle spaces of closed orbits ........................ 182
13 Examples .................................................. 185
13.1 Cycle spaces of open SL(n; R)-orbits ................. 186
13.2 Cycle spaces for open SU(p, q; F)-orbits ............. 190
13.3 Slice methods and trace transforms for SU(2, 1)
domains .............................................. 198
Part III Analytic and Geometric Consequences
14 The Double Fibration Transform ............................ 207
14.1 Double fibration ..................................... 208
14.2 Pullback ............................................. 208
14.3 Pushdown ............................................. 209
14.4 Local G-structure of G0-bundles ...................... 210
14.5 The Schubert fibration ............................... 211
14.6 Contractibility of the fiber ......................... 213
14.7 Unitary representations of real reductive Lie
groups ............................................... 214
15 Variation of Hodge Structure .............................. 217
16 Cycles in the K3 Period Domain ............................ 225
16.1 Position of K3 surfaces in the Kodaira
classification ....................................... 226
16.2 Three classes of examples ............................ 227
16.3 Parameterizing K3 surfaces ........................... 231
16.4 The cycle space MD+ ................................... 234
Part IV The Full Cycle Space
17 Combinatorics of Normal Bundles of Base Cycles ............ 243
17.1 Characterization of compact K-orbits ................. 243
17.2 Base cycles and the arrangement of Borel subgroups ... 244
17.3 Normal bundles of base cycles ........................ 245
17.4 Module structure of the tangent space of
a symmetric space .................................... 247
17.5 Shift of degree in the cohomology .................... 253
17.6 Equivariant nitrations ............................... 257
18 Methods for Computing Hl (C; O(E((q + θq)s))) .............. 259
18.1 Guide lo the computation ............................. 259
18.2 Root systems and involutions ......................... 260
18.3 Various Weyl groups .................................. 267
18.4 Some distinguished weights ........................... 271
18.5 Computation of Bott-regular weights .................. 277
18.6 Algorithm for computing the module structure of
T|C|C(Z) ............................................. 284
19 Classification for Simple g0 with rank f < rank g ......... 289
19.1 Strategy ............................................ 290
19.2 The series for g0 = sl(2r; R) ........................ 290
19.3 The series for g0 = sl(2r + 1; R) .................... 293
19.4 The series for g0 = so(2p + 1, 2q + 1) ............... 296
19.5 The case g0 = ε6C4 ................................... 302
19.6 Preliminaries for the cases where g0 is of complex
type ................................................ 304
19.7 The series for g0 = so(2r + 1; C) .................... 305
19.8 The series for g0 = sp(r; C) ......................... 306
19.9 The case g0 = f4(C) .................................. 306
19.10 The case g0 = g2(C) .................................. 306
20 Classification for rank f = rank g ........................ 309
20.1 The series for g0 = sp(r; R) .......................... 309
20.2 The series for g0 = so(2p, 2q + 1) .................... 313
20.3 The case g0 = f4,C3A1 .................................. 317
20.4 The case g0 = g2,A1A1 .................................. 319
20.5 Final table .......................................... 321
References .................................................... 323
Index ......................................................... 331
Symbol Index .................................................. 337
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