Chapter 1. Introduction ........................................ 1
1.1 Differential operators on the isotropic cone ............... 3
1.2 'Fourier transform'  C on the isotropic cone C ............. 5
1.3 Kernel of C and Bessel distributions ...................... 9
1.4 Perspectives from representation theory - finding
smallest objects .......................................... 12
1.5 Minimal representations of simple Lie groups .............. 13
1.6 Schrodinger model for the Weil representation ............. 15
1.7 Schrodinger model for the minimal representation of
O(p,q) .................................................... 16
1.8 Uncertainty relation - inner products and G-actions ....... 19
1.9 Special functions and minimal representations ............. 22
1.10 Organization of this book ................................. 25
1.11 Acknowledgements .......................................... 25
Chapter 2. Two models of the minimal representation of
O(p, q) ............................................ 27
2.1 Conformal model ........................................... 28
2.2 L2-model (the Schrodinger model) .......................... 31
2.3 Lie algebra action on L2(C) ............................... 33
2.4 Commuting differential operators on C ..................... 36
2.5 The unitary inversion operator C - π(ω0) ................. 43
Chapter 3. K-finite eigenvectors in the Schrodinger model
L2(C) .............................................. 51
3.1 Result of this chapter .................................... 51
3.2 K ∩ Mmax-invariant subspaces Hik .......................... 53
3.3 Integral formula for the K ∩ Mmax-intertwiner ............. 55
3.4 .ftT-fmite vectors ƒl,k in L2(C) .......................... 55
3.5 Proof of Theorem 3.1.1 .................................... 57
Chapter 4. Radial part of the inversion ....................... 59
4.1 Result of this chapter .................................... 59
4.2 Proof of Theorem 4.1.1 (1) ................................ 62
4.3 Preliminary results on multiplier operators ............... 63
4.4 Reduction to Fourier analysis ............................. 66
4.5 Kernel function Kl,k ...................................... 68
4.6 Proof of Theorem 4.1.1 (2) ................................ 73
Chapter 5. Main theorem ....................................... 77
5.1 Result of this chapter .................................... 77
5.2 Radon transform for the isotropic cone C .................. 79
5.3 Spectra of K'-invariant operators on Sp-2 x Sq-2 ........... 81
5.4 Proof of Theorem 5.1.1 .................................... 84
5.5 Proof of Lemma 5.4.2 (Hermitian case q = 2) ............... 85
5.6 Proof of Lemma 5.4.2 (p, q > 2) ........................... 86
Chapter 6. Bessel distributions ............................... 89
6.1 Meijer's G-distributions .................................. 89
6.2 Integral expression of Bessel distributions ............... 93
6.3 Differential equations for Bessel distributions ........... 99
Chapter 7. Appendix: special functions ....................... 105
7.1 Riesz distribution xλ+ ................................... 105
7.2 Bessel functions Jυ, Iυ, Kυ, Yυ .......................... 107
7.3 Associated Legendre functions Pυμ ........................ 111
7.4 Gegenbauer polynomials Cjμ ............................... 111
7.5 Spherical harmonics  j ( m) and branching laws ........... 113
7.6 Meijer's G-functions G ........................ 114
7.7 Appell's hypergeometric functions F1,F2,F3,F4 ............ 119
7.8 Hankel transform with trigonometric parameters ........... 120
7.9 Fractional integral of two variables ..................... 122
Bibliography .................................................. 125
List of Symbols ............................................... 129
Index ......................................................... 131
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