Paley-wiener Theorem for Line Bundles over Compact Symmetric Spaces
نویسندگان
چکیده
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2: Riemannian Symmetric Spaces and Related Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Differential Geometry, Lie Groups, and Symmetric Spaces . . . . . 5 2.2 Compact Symmetric Spaces U/K and Their Noncompact Duals G/K 8 2.3 Line Bundles over Compact Symmetric Spaces U/K . . . . . . . . . 10 2.4 Root Structures of Semisimple Lie Algebras . . . . . . . . . . . . . 13 Chapter 3: Fourier Analysis Related to Line Bundles over Compact Symmetric Spaces U/K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.1 Harmonic Analysis on Compact Groups . . . . . . . . . . . . . . . . 22 3.2 Representations of Compact Semisimple Lie Groups . . . . . . . . . 24 3.2.1 Theory of Highest Weights . . . . . . . . . . . . . . . . . . . 24 3.2.2 Spherical Representations . . . . . . . . . . . . . . . . . . . 25 3.2.3 χl-spherical Representations . . . . . . . . . . . . . . . . . . 27 3.3 Harmonic Analysis on Line Bundles over U/K . . . . . . . . . . . . 34 3.3.1 Spherical Functions of type χl on U . . . . . . . . . . . . . . 34 3.3.2 Spherical Functions of type χl on G . . . . . . . . . . . . . . 39 Chapter 4: Invariant Differential Operators . . . . . . . . . . . . . . . . . . . . . . 44 4.1 The Harish-Chandra Isomorphism . . . . . . . . . . . . . . . . . . . 44 4.2 The Hypergeometric Differential Equations . . . . . . . . . . . . . . 48 Chapter 5: The Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . 52 5.1 The Harish-Chandra Expansion . . . . . . . . . . . . . . . . . . . . 52 5.2 The Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . 54 Chapter 6: Paley-Wiener Theorem for Line Bundles over Compact Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.1 Paley-Wiener Space and Paley-Wiener Theorem . . . . . . . . . . . 63 6.2 χl-Spherical Fourier Transform Maps Into Paley-Wiener Space . . . 68 6.3 Central Functions on Compact Lie Groups . . . . . . . . . . . . . . 70 6.4 Bijectivity of χl-Spherical Fourier Transform . . . . . . . . . . . . . 72 6.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.4.2 Proof of Bijectivity of Sl . . . . . . . . . . . . . . . . . . . . 74 6.4.3 Unique Extension of Sl . . . . . . . . . . . . . . . . . . . . . 78
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