Arithmetical investigations : representation theory, orthogonal polynomials, and quantum interpolations / Shai M.J. Haran.

Format:

Book

Author/Creator:

Haran, M. J. Shai.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1941.

Lecture notes in mathematics ; 1941

Language:

English

Subjects (All):

Interpolation.

p-adic numbers.

Representations of quantum groups.

Physical Description:

xii, 217 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : Springer, [2008]

Summary:

In this book 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 Z[subscript p], which are the inverse limit of the finite rings Z/p[superscript n]. This gives rise to a tree, and probability measures w on Z[subscript p] correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L[subscript 2](Z[subscript p],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 L[subscript 2]([-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 GL[subscript n](q) that interpolates between the p-adic group GL[subscript n](Z[subscript p]), and between its real (and complex) analogue - the orthogonal O[subscript n] (and unitary U[subscript n]) 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.

Contents:

0 Introduction: Motivations from Geometry 1

0.2 Analogies Between Arithmetic and Geometry 2

0.3 Zeta Function for Curves 3

0.4 The Riemann-Roch Theorem 5

0.5 The Castelnuovo-Severi Inequality 7

0.6 Zeta Functions for Number Fields 10

0.7 Weil's Explicit Sum Formula 14

1 Gamma and Beta Measures 19

1.1 Quotients Z[subscript p]/Z*[subscript p] and P[superscript 1] (Q[subscript p])/Z*[subscript p] x Z[subscript p] 20

1.1.1 Z[subscript p]/Z*[subscript p] 20

1.1.2 P[superscript 1] (Q[subscript p])/Z*[subscript p] x Z[subscript p] 21

1.2 [gamma]-Measure on Q[subscript p] 24

1.2.1 p-[gamma]-Integral 24

1.2.2 [eta]-[gamma]-Integral 25

1.2.3 [gamma]-Measure on Q[subscript p] 25

1.3 [beta]-Measure on P[superscript 1](Q[subscript p]) 25

1.3.1 The Projective Space P[superscript 1](Q[subscript p]) 25

1.3.2 [beta]-Integral 27

1.3.3 [beta]-Measure on P[superscript 1](Q[subscript p]) 28

1.4 Remarks on the [gamma] and [beta]-Measure 29

1.4.1 [beta]-Measure Gives [gamma]-Measure 29

1.4.2 [gamma]-Measure Gives [beta]-Measure 30

1.4.3 Special Case [alpha] = [beta] = 1 31

2 Markov Chains 33

2.1 Markov Chain on Trees 34

2.1.1 Probability Measures on [partial differential]X 34

2.1.2 Hilbert Spaces 35

2.1.3 Symmetric p-Adic [beta]-Chain 36

2.1.4 Non-Symmetric p-Adic [beta]-Chain 37

2.1.5 p-Adic [gamma]-Chain 40

2.2 Markov Chain on Non-Trees 41

2.2.1 Non-Tree 41

2.2.2 Harmonic Functions 42

2.2.3 Martin Kernel 44

3 Real Beta Chain and q-Interpolation 47

3.1 Real [beta]-Chain 47

3.1.1 Probability Measure 48

3.1.2 Green Kernel and Martin Kernel 49

3.1.3 Boundary 50

3.1.4 Harmonic Measure 51

3.2 q-Interpolation 52

3.2.1 Complex [beta]-Chain 52

3.2.2 q-Zeta Functions 53

3.3 q-[beta]-Chain 55

3.3.1 q-Binomial Theorem 56

3.3.2 Probability Measure 57

3.3.3 Green Kernel and Martin Kernel 58

3.3.4 Boundary 59

3.3.5 Harmonic Measure 60

4 Ladder Structure 63

4.1 Ladder for Trees 67

4.2 Ladder for the q-[beta]-Chain 70

4.2.1 Finite Layer: The q-Hahn Basis 70

4.2.2 Boundary: The q-Jacobi Basis 74

4.3 Ladder for q-[gamma]-Chain 77

4.3.1 Finite Layer: The Finite q-Laguerre Basis 77

4.3.2 Boundary: The q-Laguerre Basis 78

4.4 Ladder for [eta]-[beta]-Chain 81

4.4.1 Finite Layer: The [eta]-Hahn Basis 81

4.4.2 Boundary: The [eta]-Jacobi Basis 82

4.5 The [eta]-Laguerre Basis 87

4.6 Real Units 89

5 q-Interpolation of Local Tate Thesis 95

5.1 Mellin Transforms 98

5.1.1 Classical Cases 98

5.1.2 q-Interpolations 103

5.2 Fourier-Bessel Transforms 106

5.2.1 Fourier Transform on H[superscript beta subscript p] 106

5.2.2 q-Fourier Transform 107

5.2.3 Convolutions 109

5.3 The Basic Basis 111

6 Pure Basis and Semi-Group 117

6.1 The Pure Basis 118

6.2 The Semi-Group G[superscript beta] 121

6.3 Global Tate-Iwasawa Theory 125

7 Higher Dimensional Theory 131

7.1 Higher Dimensional Cases 132

7.1.1 q-[beta]-Chain 132

7.1.2 The p-Adic Limit of the q-[beta]-Chain 136

7.1.3 The Real Limit of the q-[beta]-Chain 136

7.2 Representations of GL[subscript d](Z[subscript p]), p [greater than or equal] [eta], on Rank-1 Symmetric Spaces 137

8 Real Grassmann Manifold 143

8.1 Measures on the Higher Rank Spaces 143

8.1.1 Grassmann Manifolds 143

8.1.2 Measures on O[subscript m], X[superscript d subscript m] and V[superscript d subscript m] 145

8.1.3 Measures on [Omega subscript m] 148

8.2 Explicit Calculations 148

8.2.1 Measures 148

8.2.2 Metrics 151

8.3 Higher Rank Orthogonal Polynomials 153

8.3.1 Real Case 153

8.3.2 General Case 155

9 p-Adic Grassmann Manifold 157

9.1 Representation of GL[subscript d](Z[subscript p]) 157

9.1.1 Measures on GL[subscript d](Z[subscript p]), V[superscript d subscript m] and X[superscript d subscript m] 157

9.1.2 Unitary Representations of GL[subscript d](Z[subscript p]) and G[subscript N superscript d] 160

9.2 Harmonic Measure 164

9.2.1 Notations 164

9.2.2 Harmonic Measure on [Omega superscript d subscript m] 165

9.3 Basis for the Hecke Algebra 169

10 q-Grassmann Manifold 173

10.1 q-Selberg Measures 173

10.1.1 The p-Adic Limit of the q-Selberg Measures 174

10.1.2 The Real Limit of the q-Selberg Measures 175

10.2 Higher Rank q-Jacobi Basis 176

10.3 Quantum Groups 178

10.3.1 Higher Rank Quantum Groups 178

10.3.2 The Universal Enveloping Algebra 181

10.3.3 Quantum Grassmann Manifolds 182

11 Quantum Group U[subscript q](su(1,1)) and the q-Hahn Basis 185

11.1 The Quantum Universal Enveloping Algebra U[subscript q](su(1,1)) 185

11.1.1 Deformation of U(sl(2, C)) 185

11.1.2 The [beta]-Highest Weight Representation 187

11.1.3 Limits of the Subalgebras U[superscript plus or minus subscript q] 189

11.1.4 The Hopf Algebra Structure 190

11.2 Tensor Product Representation 193

11.3 The Universal R-Matrix 196

A Problems and Questions 199

B Orthogonal Polynomials 203.

Notes:

Includes bibliographical references and index.

ISBN:

9783540783787

3540783784

3540783792

9783540783794

OCLC:

220011136