The cohomology groups of locally symmetric spaces form a crucible in which many spectacular results in number theory have been forged. The ideal situation is when the locally symmetric space is the complex points of a quasi-projective scheme over a number field. In this setting the cohomology groups can be interpreted arithmetically in terms of Galois representations using etale cohomology, and interpreted analytically in terms of automorphic representations with nonzero \((\mathfrak{g},K)\)-cohomology.
On the other hand, there are automorphic representations that should be linked to Galois representations but whose \((\mathfrak{g},K)\)-cohomology vanishes. In a provocative series of papers Carayol has suggested the possibility of arithmetic interpretations of some of these automorphic representations, not in terms of the cohomology of locally symmetric spaces, but in terms of the cohomology of closely related period domains. Motivated by this conjectural link to arithmetic, this TRT is dedicated to the study of the cohomology of period domains.
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