Memoirs of the American Mathematical Society 2011; 132 pp; softcover Volume: 213 ISBN10: 0821847570 ISBN13: 9780821847572 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. 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
