eBook ISBN:  9781470406172 
Product Code:  MEMO/213/1000.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470406172 
Product Code:  MEMO/213/1000.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 213; 2011; 132 ppMSC: Primary 22; Secondary 43
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

Chapters

1. Introduction

2. Two models of the minimal representation of $O(p,q)$

3. $K$finite eigenvectors in the Schrödinger model $L^2(C)$

4. Radial part of the inversion

5. Main theorem

6. Bessel distributions

7. Appendix: special functions


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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.

Chapters

1. Introduction

2. Two models of the minimal representation of $O(p,q)$

3. $K$finite eigenvectors in the Schrödinger model $L^2(C)$

4. Radial part of the inversion

5. Main theorem

6. Bessel distributions

7. Appendix: special functions