Cycle Spaces of Flag Domains

by Fels, Gregor; Huckleberry, Alan T.; Wolf, Joseph A.

• ISBN13: 9780817643911
• ISBN10: 0817643915
• Format: Hardcover
• Copyright: 2005-12-12
• Publisher: Birkhauser

Summary

This monograph, divided into four parts, presents a comprehensive treatment and systematic examination of cycle spaces of flag domains. Assuming only a basic familiarity with the concepts of Lie theory and geometry, this work presents a complete structure theory for these cycle spaces, as well as their applications to harmonic analysis and algebraic geometry.Key features:* Accessible to readers from a wide range of fields, with all the necessary background material provided for the nonspecialist* Many new results presented for the first time* Driven by numerous examples* The exposition is presented from the complex geometric viewpoint, but the methods, applications and much of the motivation also come from real and complex algebraic groups and their representations, as well as other areas of geometry* Comparisons with classical Barlet cycle spaces are given* Good bibliography and index.Researchers and graduate students in differential geometry, complex analysis, harmonic analysis, representation theory, transformation groups, algebraic geometry, and areas of global geometric analysis will benefit from this work.

Table of Contents

Introduction	xi
Part I Introduction to Flag Domain Theory
Structure of Complex Flag Manifolds	5	(22)
Structure theory and the root decomposition	5	(4)
Cartan highest weight theory	9	(5)
Borel subgroups and subalgebras	14	(3)
Parabolic subgroups and subalgebras	17	(3)
Homogeneous holomorphic vector bundles	20	(2)
The Bott--Borel--Weil Theorem	22	(5)
Real Group Orbits	27	(4)
Bruhat Lemma and an application	27	(1)
Real isotropy	28	(3)
Orbit Structure for Hermitian Symmetric Spaces	31	(6)
Strongly orthogonal roots	31	(2)
Orbits and Cayley transforms	33	(4)
Open Orbits	37	(18)
Automorphisms and regular elements	37	(1)
Fundamental Cartan subalgebras and open G0-orbits	38	(2)
Compact subvarieties of open orbits	40	(2)
Holomorphic functions	42	(2)
Measurability of open orbits	44	(2)
Background on Levi geometry	46	(4)
The exhaustion function for measurable open orbits	50	(5)
The Cycle Space of a Flag Domain	55	(22)
Definitions and first properties	55	(2)
The three cases	57	(1)
Cycle spaces of measurable open orbits are Stein	58	(2)
The cycle space in the hermitian case	60	(6)
The classical hermitian case	66	(11)
Part II Cycle Spaces as Universal Domains
Universal Domains	77	(16)
Definitions and first properties	78	(5)
Adapted structure	83	(1)
Invariant C R structure and pseudoconvexity	84	(9)
B-Invariant Hypersurfaces in MZ	93	(20)
Iwasawa--Borel subgroups and their Schubert varieties	94	(2)
Envelope construction	96	(5)
Schubert intersection properties	101	(3)
Trace transform	104	(9)
Orbit Duality via Momentum Geometry	113	(12)
Coadjoint orbits	114	(2)
The K0-energy function	116	(5)
Duality	121	(2)
Orbit ordering	123	(2)
Schubert Slices in the Context of Duality	125	(8)
Schubert slices in arbitrary G0-orbits	126	(4)
Supporting hypersurfaces at the boundary of MD	130	(3)
Analysis of the Boundary of u	133	(30)
Preparation	135	(6)
Linearization and the Jordan decomposition	141	(4)
Characterization of closed orbits	145	(3)
The slice theorem and related isotropy computations	148	(4)
Example: Two-dimensional affine quadric	152	(2)
sl2 models at generic points of bd(u)	154	(9)
Invariant Kobayashi-Hyperbolic Stein Domains	163	(12)
Hyperbolicity of domains in G/K	164	(5)
The maximal invariant Kobayashi-hyperbolic Stein domain in an sl2-model	169	(2)
Maximality and the characterization of cycle domains	171	(4)
Cycle Spaces of Lower-Dimensional Orbits	175	(10)
Definition of cycle space	176	(1)
Intersection with Schubert varieties	177	(3)
Hypersurfaces in the complement of the cycle space	180	(1)
Cycle Spaces of nonclosed orbits	181	(1)
Cycle spaces of closed orbits	182	(3)
Examples	185	(22)
Cycle spaces of open SL(n; R)-orbits	186	(4)
Cycle spaces for open SU(p, q; F)-orbits	190	(8)
Slice methods and trace transforms for SU(2, 1) domains	198	(9)
Part III Analytic and Geometric Consequences
The Double Fibration Transform	207	(10)
Double fibration	208	(1)
Pullback	208	(1)
Pushdown	209	(1)
Local G-structure of G0-bundles	210	(1)
The Schubert fibration	211	(2)
Contractibility of the fiber	213	(1)
Unitary representations of real reductive Lie groups	214	(3)
Variation of Hodge Structure	217	(8)
Cycles in the K3 Period Domain	225	(18)
Position of K3 surfaces in the Kodaira classification	226	(1)
Three classes of examples	227	(4)
Parameterizing K3 surfaces	231	(3)
The cycle space MD+	234	(9)
Part IV The Full Cycle Space
Combinatorics of Normal Bundles of Base Cycles	243	(16)
Characterization of compact K -orbits	243	(1)
Base cycles and the arrangement of Borel subgroups	244	(1)
Normal bundles of base cycles	245	(2)
Module structure of the tangent space of a symmetric space	247	(6)
Shift of degree in the cohomology	253	(4)
Equivariant filtrations	257	(2)
Methods for Computing H1(C; O(E((q + θq)s)))	259	(30)
Guide to the computation	259	(1)
Root systems and involutions	260	(7)
Various Weyl groups	267	(4)
Some distinguished weights	271	(6)
Computation of Bott-regular weights	277	(7)
Algorithm for computing the module structure of T[C]C(Z)	284	(5)
Classification for Simple go with rank l < rank g	289	(20)
Strategy	290	(1)
The series for g0 = sl(2r; R)	290	(3)
The series for g0 = sl(2r + 1; R)	293	(3)
The series for g0 = so(2p + 1, 2q + 1)	296	(6)
The case g0 = e6,C4	302	(2)
Preliminaries for the cases where go is of complex type	304	(1)
The series for g0 = so(2r + 1; C)	305	(1)
The series for g0 = sp(r; C)	306	(1)
The case g0 = f4(C)	306	(1)
The case g0 = g2(C)	306	(3)
Classification for rank l = rank g	309	(14)
The series for g0 = sp(r; R)	309	(4)
The series for g0 = so(2p, 2q + 1)	313	(4)
The case g0 = f4,C3A1	317	(2)
The case g0 = g2,A1A1	319	(2)
Final table	321	(2)
References	323	(8)
Index	331	(6)
Symbol Index	337