2 INTRODUCTION We have attempted to keep the lecture notes as accessible as possible. Both the subjects of Hodge theory and representation theory are highly developed and extensive areas of mathematics and we are only able to touch on some aspects where they are related. When more advanced concepts from another area have been used, such as local cohomology and Grothendieck duality from algebraic geometry at the end of Lecture 6, we have illustrated them through the running examples in the hope that at least the flavor of what is being done will come through. Lectures 1 and 2 are basically elementary, assuming some standard Riemann surface theory. In this setting we will introduce essentially all of the basic concepts that appear later. Their purpose is to present up front the main ideas in the theory, both for reference and to try to give the reader a sense of what is to come. At the end of Lecture 2 we have given a more extensive summary of the topics that are covered in the later lectures and in the appendices. The reader may wish to use this as a more comprehensive introduction. Lecture 3 is essentially self-contatined, although some terminology from Lie theory and algebraic groups will be used. Lecture 4 will draw on the structure and representation theory of complex Lie algebras and their real forms. Lecture 5 will use some of the basic material about infinite dimensional representation theory and the theory of homogeneous complex manifolds. In Lectures 6 and 7 we will draw from complex function theory and, in the last part of Lecture 6, some topics from algebraic geometry. Lectures 8 and 9 will utilize the material that has gone before they are mainly devoted to specific computations in the framework that has been established. The final Lecture 10 is devoted to issues and questions that arise from the earlier lectures. We refer to the end of Lecture 2 for a more detailed account of the contents of the lectures and appendices. As selected general references to the topics covered in this work we mention for a general theory of complex manifolds, [Cat1], [Ba], [De], [GH], [Huy] and [We] for Hodge theory, in addition to the above references, [Cat2], [ET], [PS], [Vo1], [Vo2] for period domains and variations of Hodge structure, in addition to the refer- ences just listed, [CM-SP], [Ca] for Mumford-Tate groups and domains and Hodge representation [Mo1], [Mo2], [GGK1] and [Ro1] for general references for Lie groups [Kn1] and for representation theory [Kn2] specific references for topics covered in Lecture 5 are the expository papers [Sch2], [Sch3] for a general reference for flag varieties and flag domains [FHW] [GS1] for an early treatment of some of the material presented below, and [GGK2], [GG1] and [GG2] for a more extensive discussion of some of the topics covered in this monograph for a general reference for Penrose transforms [BE] and [EGW] [GGK2], [GG1] for the material in this work for mixed Hodge structures [PS] and [ET], for limiting mixed Hodge structures [CKS1], [CKS2], and [KU], [KP1] and [KP2] for boundary components of Mumford-Tate domains for the classical theory of Shimura varieties from a Hodge-theoretic perspective [Ke2].
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