Intro
Preface
Contents
1 Introduction
1.1 Admissible Banach Space Representations
1.2 Principal Series Representations
1.3 Some Questions and Further Reading
1.4 Prerequisites
1.5 Notation
1.6 Groups
Part I Banach Space Representations of p-adic Lie Groups
2 Iwasawa Algebras
2.1 Projective Limits
2.1.1 Universal Property of Projective Limits
2.1.2 Projective Limit Topology
Cofinal Subsystem
Morphisms of Inverse Systems
2.2 Projective Limits of Topological Groups and oK-Modules
2.2.1 Profinite Groups
Topology on Profinite Groups

2.3 Iwasawa Rings
2.3.1 Linear-Topological oK-Modules
Definition of Iwasawa Algebra
Fundamental System of Neighborhoods of Zero
Embedding oK[G0], G0, and oK into oK[[G0]]
2.3.2 Another Projective Limit Realization of oK[[G0]]
2.3.3 Some Properties of Iwasawa Algebras
Zero Divisors
Augmentation Map
Iwasawa Algebra of a Subgroup
3 Distributions
3.1 Locally Convex Vector Spaces
3.1.1 Banach Spaces
3.1.2 Continuous Linear Operators
3.1.3 Examples of Banach Spaces
Banach Space of Bounded Functions
Continuous Functions on G0
Mahler Expansion

3.1.4 Double Duals of a Banach Space
3.2 Distributions
3.2.1 The Weak Topology on Dc(G0,oK)
3.2.2 Distributions and Iwasawa Rings
3.2.3 The Canonical Pairing
3.3 The Bounded-Weak Topology
3.3.1 The Bounded-Weak Topology is Strictly Finer than the Weak Topology
The Weak Topology on V'
The Bounded-Weak Topology on V'
3.4 Locally Convex Topology on K[[G0]]
3.4.1 The Canonical Pairing
3.4.2 p-adic Haar Measure
3.4.3 The Ring Structure on Dc(G0,K)
A Big Projective Limit
4 Banach Space Representations
4.1 p-adic Lie Groups

4.2 Linear Operators on Banach Spaces
4.2.1 Spherically Complete Spaces
4.2.2 Some Fundamental Theorems in Functional Analysis
4.2.3 Banach Space Representations: Definition and Basic Properties
4.3 Schneider-Teitelbaum Duality
4.3.1 Schikhof's Duality
4.3.2 Duality for Banach Space Representations: Iwasawa Modules
K[[G0]]-module structure on V'
4.4 Admissible Banach Space Representations
4.4.1 Locally Analytic Vectors: Representations in Characteristic p
Locally Analytic Vectors
Unitary Representations and Reduction Modulo pK

4.4.2 Duality for p-adic Lie Groups
Part II Principal Series Representations of Reductive Groups
Notation in Part II
5 Reductive Groups
5.1 Linear Algebraic Groups
5.1.1 Basic Properties of Linear Algebraic Groups
More Examples of Linear Algebraic Groups
Unipotent Subgroups
Identity Component
Tori
5.1.2 Lie Algebra of an Algebraic Group
Lie Algebras
Lie Algebra of an Algebraic Group
5.2 Reductive Groups Over Algebraically Closed Fields
5.2.1 Rational Characters
5.2.2 Roots of a Reductive Group
Weyl Group
Abstract Root Systems
Simple Roots