Contents Introduction 1 Acknowledgement 3 Lecture 1. The Classical Theory: Part I 5 Beginnings of representation theory1 13 Lecture 2. The Classical Theory: Part II 17 Lecture 3. Polarized Hodge Structures and Mumford-Tate Groups and Domains 31 Lecture 4. Hodge Representations and Hodge Domains 51 Lecture 5. Discrete Series and n-Cohomology 69 Introduction 69 Appendix to Lecture 5: The Borel-Weil-Bott (BWB) theorem 91 Lecture 6. Geometry of Flag Domains: Part I 95 Appendix to Lecture 6: The GR- and KC- orbit structure of ˇ and the GR-orbit structure of U 120 Lecture 7. Geometry of Flag Domains: Part II 147 Appendix to Lecture 7: The Borel-Weil-Bott theorem revisited 161 Lecture 8. Penrose Transforms in the Two Main Examples 165 Appendix to Lecture 8: Proofs of the results on Penrose transforms for D and D 178 Lecture 9. Automorphic Cohomology 191 Appendix I to Lecture 9: The K-types of the TDLDS for SU(2, 1) and Sp(4) 209 Appendix II to Lecture 9: Schmid’s proof of the degeneracy of the HSSS for TDLDS in the SU(2, 1) and Sp(4) cases 214 Appendix III to Lecture 9: A general result relating TDLDS and Dolbeault cohomology of Mumford-Tate domains 218 Lecture 10. Miscellaneous Topics and Some Open Questions 221 Appendix to Lecture 10: Boundary components and Carayol’s result 245 Bibliography 299 Index 303 iii

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