Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations, Issue 1941

Front Cover
Springer Science & Business Media, Apr 25, 2008 - Mathematics - 217 pages
0 Reviews

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.

 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Selected pages

Contents

Introduction Motivations from Geometry
1
02 Analogies Between Arithmetic and Geometry
2
03 Zeta Function for Curves
3
04 The RiemannRoch Theorem
5
05 The CastelnuovoSeveri Inequality
7
06 Zeta Functions for Number Fields
10
07 Weils Explicit Sum Formula
14
Gamma and Beta Measures
19
45 The 𝜂Laguerre Basis
87
46 Real Units
89
qInterpolation of Local Tate Thesis
95
51 Mellin Transforms
98
512 qInterpolations
103
52 FourierBessel Transforms 521 Fourier Transform on H𝜷p
106
522 qFourier Transform
107
523 Convolutions
109

11 Quotients ℤpℤp and ℙℚpℤp x ℤp
20
ℙℚpℤp x ℤp
21
12 𝜸Measure on ℚp
24
122 𝜂𝜸Integral
25
132 𝜷Integral
27
133 𝜷Measure on ℙℚp
28
14 Remarks on the 𝜸 and 𝜷Measure
29
142 𝜸Measure Gives 𝜷Measure
30
143 Special Case 𝜶 𝜷 1
31
Markov Chains
32
21 Markov Chain on Trees
34
212 Hilbert Spaces
35
213 Symmetric pAdic 𝜷Chain
36
214 NonSymmetric pAdic 𝜷Chain
37
215 pAdic 𝜸Chain
40
22 Markov Chain on NonTrees
41
222 Harmonic Functions
42
223 Martin Kernel
44
Real Beta Chain and qInterpolation
47
311 Probability Measure
48
312 Green Kernel and Martin Kernel
49
313 Boundary
50
314 Harmonic Measure
51
32 qInterpolation
52
322 qZeta Functions
53
33 q𝜷Chain
55
331 qBinomial Theorem
56
332 Probability Measure
57
333 Green Kernel and Martin Kernel
58
334 Boundary
59
335 Harmonic Measure
60
Ladder Structure
63
41 Ladder for Trees
67
42 Ladder for the q𝜷Chain
70
The qJacobi Basis
74
43 Ladder for q𝜸Chain
77
The qLaguerre Basis
78
44 Ladder for 𝜂𝜷Chain
81
The 𝜂Jacobi Basis
82
53 The Basic Basis
111
Pure Basis and SemiGroup
116
61 The Pure Basis
118
62 The SemiGroup G𝜷
121
63 Global TateIwasawa Theory
125
Higher Dimensional Theory
131
71 Higher Dimensional Cases 711 q𝜷Chain
132
712 The pAdic Limit of the q𝜷Chain
136
72 Representations of GLdℤp p 𝜂 on Rank1 Symmetric Spaces
137
Real Grassmann Manifold
143
812 Measures on Om1 Xdm and Vdm
145
813 Measures on 𝜴m
148
822 Metrics
151
83 Higher Rank Orthogonal Polynomials
153
832 General Case
155
pAdic Grassmann Manifold
157
912 Unitary Representations of GLdℤp and GN₄
160
92 Harmonic Measure
164
922 Harmonic Measure on 𝜴dm
165
93 Basis for the Hecke Algebra
169
qGrassmann Manifold
173
1011 The pAdic Limit of the qSelberg Measures
174
1012 The Real Limit of the qSelberg Measures
175
102 Higher Rank qJacobi Basis
176
103 Quantum Group
178
1032 The Universal Enveloping Algebra
181
1033 Quantum Grassmann Manifolds
182
Quantum Group Uq𝔰𝔲11 and the qHahn Basis
185
1112 The 𝜷Highest Weight Representation
187
1113 Limits of the Subalgebras Uq
189
1114 The Hopf Algebra Structure
190
112 Tensor Product Representation
193
113 The Universal 𝓡Matrix
196
Problems and Questions
198
Orthogonal Polynomials
203
Bibliography
209
Index
214
Copyright

Other editions - View all

Common terms and phrases