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

D 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