Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations

Shai M. J. Haran (eds.)3540783784, 978-3-540-78378-7

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 Zpwhich 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.

Table of contents :
Front Matter….Pages I-XII
Introduction: Motivations from Geometry….Pages 1-18
Gamma and Beta Measures….Pages 19-31
Markov Chains….Pages 33-46
Real Beta Chain and q -Interpolation….Pages 47-62
Ladder Structure….Pages 63-93
q -Interpolation of Local Tate Thesis….Pages 95-115
Pure Basis and Semi-Group….Pages 117-130
Higher Dimensional Theory….Pages 131-142
Real Grassmann Manifold….Pages 143-156
p -Adic Grassmann Manifold….Pages 157-171
q -Grassmann Manifold….Pages 173-184
Quantum Group U q (su(1, 1)) and the q -Hahn Basis….Pages 185-197
Back Matter….Pages 199-222