Arithmetical Investigations

Representation Theory, Orthogonal Polynomials, and Quantum Interpolations

  • Shai M. J. Haran

Part of the Lecture Notes in Mathematics book series (LNM, volume 1941)

Table of contents

About this book

Introduction

In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to a tree, and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp,w). The real analogue of the p-adic integers is the interval [-1,1], and a probability measure w on it gives rise to a special basis for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For special (gamma and beta) measures there is a "quantum" or "q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GLn(q)that interpolates between the p-adic group GLn(Zp), and between its real (and complex) analogue -the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums.

Keywords

Arithmetic geometry Beta DEX Finite Fourier transform Markov chain Markov chains approximation arithmetic manifold probability probability measure quantum groups representation theory special funtions

Editors and affiliations

  • Shai M. J. Haran
    • 1
  1. 1.Department of MathematicsTechnion – Israel Institute of TechnologyHaifaIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-78379-4
  • Copyright Information Springer Berlin Heidelberg 2008
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-78378-7
  • Online ISBN 978-3-540-78379-4
  • Series Print ISSN 0075-8434
  • About this book