P-adic banach space representations : with applications to principal series / / Dubravka Ban

• Autore: Ban Dubravka
• Pubblicazione: Cham, Switzerland : , : Springer Nature Switzerland AG, , [2022]; ©2022
• Edizione: 1st ed. 2022.
• Descrizione fisica: 1 online resource (219 pages)
• Disciplina: 515.732
• Soggetto topico: Banach spaces; p-adic analysis; Espais de Banach; Anàlisi p-àdica
• Soggetto genere / forma: Llibres electrònics
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: 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 -- 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.
• Sommario/riassunto: 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.
• Titolo autorizzato: P-Adic Banach Space Representations
• ISBN: 9783031226847; 9783031226830
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996511863103316
• Lo trovi qui: Univ. di Salerno
• Serie: Lecture Notes in Mathematics, . 1617-9692 ; ; 2325