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

Description

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. 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