E-grāmata: p-adic Banach Space Representations: With Applications to Principal Series

Chapter
1. Iwasawa algebras: The purpose of the chapter is to define Iwasawa algebras and study their properties. As a preparation, we first cover projective limits of topological spaces, finite groups, and linear-topological modules. After that, we explain in detail Iwasawa algebras and their topology.

Chapter
2. Distributions: We review basic definitions and properties of locally convex vector spaces. We study the algebra of continuous distributions and establish an isomorphism with the corresponding Iwasawa algebra. We discuss different topologies on the algebra of continuous distributions, among them the weak topology and the bounded-weak topology.

Chapter
3. Banach space representations: We prove some fundamental theorems in nonarchimedean functional analysis and introduce Banach space representations. We give an overview of the Schikhof duality between p-adic Banach spaces and compactoids. Then, we present the theory of admissible Banach space representations by Schneider and Teitelbaum and their duality theory.

Part II: Principal series representations of reductive groups

Chapter
4. Reductive Groups: In this chapter, we give an overview of the structure theory of split reductive Z-groups, with no proofs. The purpose of this chapter is to help a learner navigate through the literature and to explain different objects we need in
Chapters 6 and 7, such as roots, unipotent subgroups, and Iwahori subgroups. We also review important structural results, such as Bruhat decomposition, Iwasawa decomposition, and Iwahori factorization.

Chapter
5. Algebraic and smooth representations: In our study of Banach space representations, we also encounter algebraic and smooth representations. Namely, continuous principal series representations may contain finite dimensional algebraic representations or smooth principal series representations. In this chapter, we review some basic properties of these representations.

Chapter
6. Continuous principal series: We establish some basic properties of the continuous principal series representations. In particular, we prove that they are Banach. After that, we work on the dual side and study the corresponding Iwasawa modules.

Chapter
7. Intertwining operators: In this chapter, we present the main results and proofs from a recent joint work with Joseph Hundley. The purpose is to describe the space of continuous intertwining operators between principal series representations. As before, we apply the Schneider-Teitelbaum duality and work with the corresponding Iwasawa modules.