TY  - BOOK
AU  - Haran,Shai M.J.
ED  - SpringerLink (Online service)
TI  - Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations 
T2  - Lecture Notes in Mathematics,
SN  - 9783540783794
AV  - QA241-247.5 
U1  - 512.7 23
PY  - 2008///
CY  - Berlin, Heidelberg
PB  - Springer Berlin Heidelberg
KW  - Mathematics
KW  - Number theory
KW  - Number Theory
N1  - Introduction: Motivations from Geometry -- Gamma and Beta Measures -- Markov Chains -- Real Beta Chain and q-Interpolation -- Ladder Structure -- q-Interpolation of Local Tate Thesis -- Pure Basis and Semi-Group -- Higher Dimensional Theory -- Real Grassmann Manifold -- p-Adic Grassmann Manifold -- q-Grassmann Manifold -- Quantum Group Uq(su(1, 1)) and the q-Hahn Basis
N2  - 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
UR  - http://dx.doi.org/10.1007/978-3-540-78379-4
ER  -