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The Schrödinger Model for the Minimal Representation of the Indefinite Orthogonal Group \(O(p,q)\)
Toshiyuki Kobayashi, University of Tokyo, Japan, and Gen Mano, PricewaterhouseCoopers Aarata, Tokyo, Japan
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Memoirs of the American Mathematical Society
2011; 132 pp; softcover
Volume: 213
ISBN-10: 0-8218-4757-0
ISBN-13: 978-0-8218-4757-2
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/213/1000
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The authors introduce a generalization of the Fourier transform, denoted by \(\mathcal{F}_C\), on the isotropic cone \(C\) associated to an indefinite quadratic form of signature \((n_1,n_2)\) on \(\mathbb{R}^n\) (\(n=n_1+n_2\): even). This transform is in some sense the unique and natural unitary operator on \(L^2(C)\), as is the case with the Euclidean Fourier transform \(\mathcal{F}_{\mathbb{R}^n}\) on \(L^2(\mathbb{R}^n)\). Inspired by recent developments of algebraic representation theory of reductive groups, the authors shed new light on classical analysis on the one hand, and give the global formulas for the \(L^2\)-model of the minimal representation of the simple Lie group \(G=O(n_1+1,n_2+1)\) on the other hand.

Table of Contents

  • Introduction
  • Two models of the minimal representation of \(O(p,q)\)
  • \(K\)-finite eigenvectors in the Schrödinger model \(L^2(C)\)
  • Radial part of the inversion
  • Main theorem
  • Bessel distributions
  • Appendix: special functions
  • Bibliography
  • List of Symbols
  • Index
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