This book systematically develops the theory of continuous representations on p-adic Banach spaces. Its purpose is to lay the foundations of the representation theory of reductive p-adic groups on p-adic Banach spaces, explain the duality theory of Schneider and Teitelbaum, and demonstrate its applications to continuous principal series. Written to be accessible to graduate students, the book gives a comprehensive introduction to the necessary tools, including Iwasawa algebras, p-adic measures and distributions, p-adic functional analysis, reductive groups, and smooth and algebraic representations. Part 1 culminates with the duality between Banach space representations and Iwasawa modules. This duality is applied in Part 2 for studying the intertwining operators and reducibility of the continuous principal series on p-adic Banach spaces.
This monograph is intended to serve both as a reference book and as an introductory text for graduate students and researchers entering the area.
Author(s): Dubravka Ban
Series: Lecture Notes in Mathematics, 2325
Publisher: Springer
Year: 2023
Language: English
Pages: 218
City: Cham
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
5.2.3 Classification of Irreducible Root Systems
5.2.4 Classification of Reductive Groups
Cocharacters
Root Datum of a Reductive Group
Abstract Root Datum
5.2.5 Structure of Reductive Groups
Root Subgroups
Borel Subgroups and Parabolic Subgroups
5.3 F-Reductive Groups
5.4 Z-Groups
5.4.1 Algebraic R-Groups
5.4.2 Split Z-Groups
Root Subgroups
5.5 The Structure of G(L)
5.5.1 oL-Points of Algebraic Z-Groups
5.5.2 oL-Points of Split Z-Groups
5.5.3 Coset Representatives for G/P
5.6 General Linear Groups
6 Algebraic and Smooth Representations
6.1 Algebraic Representations
6.1.1 Definition and Basic Properties
6.1.2 Classification of Simple Modules of Reductive Groups
Abstract Weights
Weights of a Reductive Group
Dominant Bases of X(T)
Weights of a Module
Algebraic Induction
Simple Modules
6.2 Smooth Representations
6.2.1 Absolute Value
6.2.2 Smooth Representations and Characters
6.2.3 Basic Properties
Isomorphic Fields
Absolutely Irreducible Representations
Contragredient
Tensor Product of Representations
6.2.4 Admissible-Smooth Representations
6.2.5 Smooth Principal Series
Normalized Induction
Composition Factors of Principal Series
6.2.6 Smooth Principal Series of GL2(L) and SL2(L)
7 Continuous Principal Series
7.1 Continuous Principal Series Are Banach
7.1.1 Direct Sum Decomposition of IndP0G0(χ0-1)
7.1.2 Unitary Principal Series
7.1.3 Algebraic and Smooth Vectors
Algebraic Characters
Smooth Characters
7.1.4 Unitary Principal Series of GL2(Qp)
7.2 Duals of Principal Series
7.2.1 Module M0(χ)
7.3 Projective Limit Realization of M0(χ)
7.4 Direct Sum Decomposition of M(χ)
7.4.1 The Case G0=GL2(Zp)
7.4.2 General Case
8 Intertwining Operators
8.1 Invariant Distributions
8.1.1 Invariant Distributions on Vector Groups
8.1.2 ``Partially Invariant'' Distributions on Unipotent Groups
8.1.3 T0-Equivariant Distributions on Unipotent Groups
8.2 Intertwining Algebra
8.2.1 Ordinary Representations of GL2(Qp)
8.3 Finite Dimensional G0-Invariant Subspaces
8.3.1 Induction from the Trivial Character: Intertwiners
8.4 Reducibility of Principal Series
8.4.1 Locally Analytic Vectors
Reducibility Question for G(Qp)
Reducibility Question for G(L)
8.4.2 A Criterion for Irreducibility
A Nonarchimedean Fields and Spaces
A.1 Ultrametric Spaces
A.2 Nonarchimedean Local Fields
A.2.1 p-Adic Numbers
A.2.2 Finite Extensions of Qp
A.2.3 Algebraic Closure Qp
A.3 Normed Vector Spaces
B Affine and Projective Varieties
B.1 Affine Varieties
B.1.1 Zariski Topology on Affine Space
B.1.2 Morphisms and Products of Affine Varieties
B.2 Projective Varieties
References
Index
