Introduction This monograph is based on ten lectures given by the second author at the CBMS sponsored conference Hodge Theory, Complex Geometric and Representation Theory that was held during June, 2012 at Texas Christian University, and on selected developments that have occured since then in the general areas covered by those lectures. The original material covered in the lectures and in the appendices is largely on joint work by the three authors. This work roughly separates into two parts. One is the lectures themselves, which appear here largely as they were given at the CBMS conference and which were circulated at that time. The other part is the appendices to the later lectures. These cover material that was either related to the lecture, such as selected further background or proofs of results presented in the lectures, or new topics that are re- lated to the lecture but have been developed since the conference. We have chosen to structure this monograph in this way because the lectures give a fairly succinct, in some places informal, account of the main subject matter. The appendices then give, in addition to some further developments, further details and proofs of several of the main results presented in the lectures. These lectures are centered around the subjects of Hodge theory and represen- tation theory and their relationship. A unifying theme is the geometry of homoge- neous complex manifolds. Finite dimensional representation theory enters in multiple ways, one of which is the use of Hodge representations to classify the possible realizations of a reduc- tive, Q-algebraic group as a Mumford-Tate group. The geometry of homogeneous complex manifolds enters through the study of Mumford-Tate domains and Hodge domains and their boundaries. It also enters through the cycle and correspondence spaces associated to Mumford-Tate domains. Running throughout is the analysis of the GR-orbit structure of flag varieties and the GR-orbit structure of the complex- ifications of symmetric spaces GR/K where K contains a compact maximal torus. Infinite dimensional representation theory and the geometry of homogeneous complex manifolds interact through the realization, due primarily to Schmid, of the Harish-Chandra modules associated to discrete series representations, especially their limits, as cohomology groups associated to homogeneous line bundles. It also enters through the work of Carayol on automorphic cohomology, which involves the Hodge theory associated to Mumford-Tate domains and to their boundary components. Throughout these lectures we have kept the “running examples” of SL2, SU(2,1), Sp(4) and SO(4, 1). Many of the general results whose proofs are not given in the lectures are easily verified in the running examples. They also serve to illustrate and make concrete the general theory. 1
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