The Schrödinger model for the minimal representation of the indefinite orthogonal group O(p, q) / Toshiyuki Kobayashi, Gen Mano

Author(s): Kobayashi, Toshiyuki, 1962-
Location:
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Subjects: Representations of Lie groups
Schrödinger equation
Formats: Print
Material Type: Books
Language: English
Audience: Unspecified
Published: Providence, R.I. : American Mathematical Society, 2011
Series: Memoirs of the American Mathematical Society no. 1000
Memoirs of the American Mathematical Society no. 1000
LC Classification: Q, QA
Physical Description: v, 132 p. : ill. ; 26 cm
Table of Contents: ch. 1 . Introduction 1
1.1.. Differential operators on the isotropic cone 3
1.2.. 'Fourier transform' FC on the isotropic cone C 5
1.3.. Kernel of FC 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.. Schrödinger model for the Weil representation 15
1.7.. Schrödinger 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
ch. 2 . Two models of the minimal representation of O(p, q) 27
2.1.. Conformal model 28
2.2.. L2-model (the Schrödinger 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 FC = π(w0) 43
ch. 3 . K-finite eigenvectors in the Schrodinger model L2(C) 51
3.1.. Result of this chapter 51
3.2.. K inter section Mmax-invariant subspaces Hl,k 53
3.3.. Integral formula for the K inter section Mmax-intertwines 55
3.4.. K-finite vectors fl,k in L2(C) 55
3.5.. Proof of Theorem 3.1.1 57
ch. 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
ch. 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
ch. 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
ch. 7 . Appendix: special functions 105
7.1.. Riesz distribution xγ+ 105
7.2.. Bessel functions Jv, Iv, Kv, Yv 107
7.3.. Associated Legendre functions Pμν 111
7.4.. Gegenbauer polynomials Cμl 111
7.5.. Spherical harmonics Hj(Rm) and branching laws 113
7.6.. Meijer's G-functions Gm,n p,q (x 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
Additional Authors: Mano, Gen, 1977-
Notes: LCCN: 2011019994
ISBN: 9780821847572 (alk. paper)
ISBN: 0821847570 (alk. paper)
"Volume 213, number 1000 (first of 5 numbers )."
Includes bibliographical references (p. 125-127) and index
OCLC Number: 728892093
ISBN/ISSN: 9780821847572
0821847570
0065-9266