Contents
Chapter 1. Introduction 1
Notations and conventions 7
Chapter 2. Oscillatory integrals 11
2.1. Symbol spaces 12
2.2. Tempered pairs 23
2.3. An oscillatory integral for admissible tempered pairs 31
2.4. A Fubini Theorem for semi-direct products 34
2.5. A Schwartz space for tempered pairs 36
2.6. Bilinear mappings from the oscillatory integral 38
Chapter 3. Tempered pairs for ahlerian Lie groups 43
3.1. Pyatetskii-Shapiro’s theory 43
3.2. Geometric structures on elementary normal j-groups 45
3.3. Tempered pair for elementary normal j-groups 48
3.4. Tempered pairs for normal j-groups 53
Chapter 4. Non-formal star-products 59
4.1. Star-products on normal j-groups 59
4.2. An oscillatory integral formula for the star-product 62
Chapter 5. Deformation of Fr´ echet algebras 67
5.1. The deformed product 67
5.2. Relation with the fixed point algebra 73
5.3. Functorial properties of the deformed product 74
Chapter 6. Quantization of polarized
symplectic symmetric spaces 79
6.1. Polarized symplectic symmetric spaces 81
6.2. Unitary representations of symmetric spaces 86
6.3. Locality and the one-point phase 90
6.4. Unitarity and midpoints for elementary spaces 91
6.5. The -product as the composition law of symbols 96
6.6. The three-point kernel 98
6.7. Extensions of polarization quadruples 101
Chapter 7. Quantization of ahlerian Lie groups 105
7.1. The transvection quadruple of an elementary normal j-group 105
7.2. Quantization of elementary normal j-groups 110
7.3. Quantization of normal j-groups 114
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