The Cohomology of GLn

This chapter addresses the cohomology of GLn. It first discusses the adèlic locally symmetric space. Next, the chapter turns to the highest weight modules 𝓜λ‎ and the sheaves 𝓜̃λ‎. From there, the chapter illustrates the cohomology of the sheaves 𝓜̃λ‎. A fundamental problem at the heart of this monograph is to understand the arithmetic information contained in the sheaf-theoretically defined cohomology groups H ⦁(SGKf,𝓜̃λ‎,E). Finally, the chapter briefly discusses how to refine many of the foregoing considerations to talk about integral sheaves and their cohomology and why this is interesting. This aspect is not so relevant for the results in this book, but it will become relevant when applying certain refinements of the results to arithmetic questions.