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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$75 Individual Members: US$45
Institutional Members: US\$60
Order Code: MEMO/213/1000

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.

• 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