By Gregor Fels
This monograph, divided into 4 components, offers a finished therapy and systematic exam of cycle areas of flag domain names. Assuming just a uncomplicated familiarity with the suggestions of Lie idea and geometry, this paintings provides an entire constitution idea for those cycle areas, in addition to their purposes to harmonic research and algebraic geometry. Key gains: * obtainable to readers from a variety of fields, with all of the precious heritage fabric supplied for the nonspecialist * Many new effects awarded for the 1st time * pushed by way of a number of examples * The exposition is gifted from the complicated geometric standpoint, however the equipment, functions and masses of the inducement additionally come from genuine and complicated algebraic teams and their representations, in addition to different parts of geometry * Comparisons with classical Barlet cycle areas are given * reliable bibliography and index. Researchers and graduate scholars in differential geometry, complicated research, harmonic research, illustration idea, transformation teams, algebraic geometry, and components of worldwide geometric research will take advantage of this paintings.
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Additional info for Cycle Spaces of Flag Domains. A Complex Geometric Viewpoint
2 says that r contains Cartan subalgebras of g. Thus j is a τ -stable Cartan subalgebra of g. We have the parabolic subgroup Q ⊂ G and the complex ﬂag manifold Z = G/Q. Since Q is its own normalizer in G, we may view Z as the space of all G-conjugates of q, by the correspondence gQ ↔ Ad(g)q. We will write qz for the parabolic subalgebra of g corresponding to z ∈ Z, and will write Qz for the corresponding parabolic subgroup of G. 2 Real isotropy Here is the principal strategy for dealing with G0 -orbits on Z.
6 The Bott–Borel–Weil Theorem Let p : Eχ → Z be an irreducible G-homogeneous holomorphic vector bundle. Write O(E) for the sheaf of (germs of) holomorphic sections. Then the natural action of G on Eχ and Z induces a representation of G on each of the Dolbeault cohomology spaces H k (Z; O(E)). The Bott–Borel–Weil Theorem gives a precise description of these cohomologies as G-modules in terms of the highest weight of χ . 6 The Bott–Borel–Weil Theorem 23 It does not matter whether one describes the H k (Z; O(E)) as G-modules or as Gu -modules, because the continuous representation theory of a compact group Gu is the same as the completely reducible holomorphic representation theory of its complexiﬁcation G.
Further, the Weyl group W (G0 , T0 ) induces every permutation of . The notion of strong orthogonality of roots was developed in various forms: by Harish-Chandra [HC1] for the study of the holomorphic discrete series, by B. Kostant [Kos1] and later by M. Sugiura [Su] for the construction and classiﬁcation of real Cartan subalgebras, and by C. C. Moore [Mo] for an analysis of compactiﬁcations of the bounded symmetric domains. 2 Orbits and Cayley transforms Let z0 = 1Q ∈ G/Q = Z denote the base point of our ﬂag manifold Z when Z is viewed as a homogeneous space.
Cycle Spaces of Flag Domains. A Complex Geometric Viewpoint by Gregor Fels