| Description |
1 online resource (190 pages) |
| Series |
Lecture notes in mathematics ; volume 2325 |
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Lecture notes in mathematics (Springer-Verlag) ; v. 2325.
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| Contents |
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 |
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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 |
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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 |
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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 |
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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 |
| Summary |
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 |
| Notes |
Print version record |
| Subject |
Banach spaces.
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p-adic analysis.
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Espacios de Banach
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Análisis p-adico
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Banach spaces
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p-adic analysis
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| Form |
Electronic book
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| ISBN |
9783031226847 |
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3031226844 |
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