p-adic Banach Space Representations

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.

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

Auteur
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.

Résumé
"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 Bosnjak, zbMATH 1523.22001, 2023)

Contenu

• 1. Introduction. - Part I Banach Space Representations of p -adic Lie Groups.
     • 2. Iwasawa Algebras. - 3. Distributions. - 4. Banach Space Representations. - Part II Principal Series Representations of Reductive Groups. -
          5. Reductive Groups. - 6. Algebraic and Smooth Representations. - 7. Continuous Principal Series. - 8. Intertwining Operators.