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p-adic Banach Space Representations

With Applications to Principal Series

  • Book
  • © 2022

Overview

Authors:
  1. Dubravka Ban

    1. School of Mathematical and Statistical Sciences, Southern Illinois University, Carbondale, USA

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  • Provides numerous exercises for graduate students and readers interested in hands-on learning
  • Offers a comprehensive introduction to the representation theory of p-adic groups on p-adic Banach spaces
  • Presents thoroughly the foundations and the Schneider-Teitelbaum duality

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2325)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Dubravka Ban
    Pages 1-7

  3. Banach Space Representations of p-adic Lie Groups

    1. Front Matter

      Pages 9-9
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    2. Iwasawa Algebras

      • Dubravka Ban
      Pages 11-34

    3. Distributions

      • Dubravka Ban
      Pages 35-61

    4. Banach Space Representations

      • Dubravka Ban
      Pages 63-87

  4. Principal Series Representations of Reductive Groups

    1. Front Matter

      Pages 89-89
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    2. Reductive Groups

      • Dubravka Ban
      Pages 91-121

    3. Algebraic and Smooth Representations

      • Dubravka Ban
      Pages 123-145

    4. Continuous Principal Series

      • Dubravka Ban
      Pages 147-172

    5. Intertwining Operators

      • Dubravka Ban
      Pages 173-192

  5. Back Matter

    Pages 193-216
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Keywords

About this book

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.


Reviews

“This is a book on the representation theory of p-adic groups on p-adic Banach spaces whose foundations were laid by Schneider and Teitelbaum. It explains their duality theory and demonstrates its applications to continuous principal series. ... It could also be of an interest to mathematicians who are working in the representation theory on complex vector spaces.” (Barbara Bošnjak, zbMATH 1523.22001, 2023)

Authors and Affiliations

  • School of Mathematical and Statistical Sciences, Southern Illinois University, Carbondale, USA

    Dubravka Ban

About the author

Dubravka Ban received her doctoral degree at the University of Zagreb. She was a postdoctoral fellow at the International Centre for Theoretical Physics in Trieste and a visiting assistant professor at Purdue University. Ban was a Humboldt research fellow at the University of Münster and University of Bonn. Currently, she is a professor of mathematics at Southern Illinois University, Carbondale. Her research is in the representation theory of p-adic groups in the context of Langlands program. Trained in the smooth representations on complex vector spaces, she is intrigued by the p-adic Banach space representations and finds them very interesting objects to study.

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