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Schrödinger Model for the Minimal Representation of the Indefinite Orthogonal Group O(p,q).

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical SocietyPublisher: Providence : American Mathematical Society, 2011Copyright date: ©2011Edition: 1st edDescription: 1 online resource (145 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470406172
Subject(s): Genre/Form: Additional physical formats: Print version:: Schrödinger Model for the Minimal Representation of the Indefinite Orthogonal Group O(p,q)DDC classification:
  • 512/.582
LOC classification:
  • QA387 -- .K633 2011eb
Online resources:
Contents:
Intro -- Contents -- Chapter 1. Introduction -- 1.1. Differential operators on the isotropic cone -- 1.2. `Fourier transform' FC on the isotropic cone C -- 1.3. Kernel of FC and Bessel distributions -- 1.4. Perspectives from representation theory - finding smallest objects -- 1.5. Minimal representations of simple Lie groups -- 1.6. Schrödinger model for the Weil representation -- 1.7. Schrödinger model for the minimal representation of O(p,q) -- 1.8. Uncertainty relation - inner products and G-actions -- 1.9. Special functions and minimal representations -- 1.10. Organization of this book -- 1.11. Acknowledgements -- Chapter 2. Two models of the minimal representation of O(p,q) -- 2.1. Conformal model -- 2.2. L2-model (the Schrödinger model) -- 2.3. Lie algebra action on L2(C) -- 2.4. Commuting differential operators on C -- 2.5. The unitary inversion operator FC=(w0) -- Chapter 3. K-finite eigenvectors in the Schrödinger model L2(C) -- 3.1. Result of this chapter -- 3.2. K Mmax-invariant subspaces Hl,k -- 3.3. Integral formula for the K Mmax-intertwiner -- 3.4. K-finite vectors fl,k in L2(C) -- 3.5. Proof of Theorem 3.1.1 -- Chapter 4. Radial part of the inversion -- 4.1. Result of this chapter -- 4.2. Proof of Theorem 4.1.1 (1) -- 4.3. Preliminary results on multiplier operators -- 4.4. Reduction to Fourier analysis -- 4.5. Kernel function Kl,k -- 4.6. Proof of Theorem 4.1.1 (2) -- Chapter 5. Main theorem -- 5.1. Result of this chapter -- 5.2. Radon transform for the isotropic cone C -- 5.3. Spectra of K'-invariant operators on Sp-2Sq-2 -- 5.4. Proof of Theorem 5.1.1 -- 5.5. Proof of Lemma 5.4.2 (Hermitian case q=2) -- 5.6. Proof of Lemma 5.4.2 (p,q&gt -- 2) -- Chapter 6. Bessel distributions -- 6.1. Meijer's G-distributions -- 6.2. Integral expression of Bessel distributions -- 6.3. Differential equations for Bessel distributions.
Chapter 7. Appendix: special functions -- 7.1. Riesz distribution x+ -- 7.2. Bessel functions J, I, K, Y -- 7.3. Associated Legendre functions P -- 7.4. Gegenbauer polynomials Cl -- 7.5. Spherical harmonics Hj(Rm) and branching laws -- 7.6. Meijer's G-functions Gp,qm,n(to1.5. x |to1.5. a1, @let@token , ap b1, @let@token , bq)to1.5. -- 7.7. Appell's hypergeometric functions F1, F2, F3, F4 -- 7.8. Hankel transform with trigonometric parameters -- 7.9. Fractional integral of two variables -- Bibliography -- List of Symbols -- Index.
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Intro -- Contents -- Chapter 1. Introduction -- 1.1. Differential operators on the isotropic cone -- 1.2. `Fourier transform' FC on the isotropic cone C -- 1.3. Kernel of FC and Bessel distributions -- 1.4. Perspectives from representation theory - finding smallest objects -- 1.5. Minimal representations of simple Lie groups -- 1.6. Schrödinger model for the Weil representation -- 1.7. Schrödinger model for the minimal representation of O(p,q) -- 1.8. Uncertainty relation - inner products and G-actions -- 1.9. Special functions and minimal representations -- 1.10. Organization of this book -- 1.11. Acknowledgements -- Chapter 2. Two models of the minimal representation of O(p,q) -- 2.1. Conformal model -- 2.2. L2-model (the Schrödinger model) -- 2.3. Lie algebra action on L2(C) -- 2.4. Commuting differential operators on C -- 2.5. The unitary inversion operator FC=(w0) -- Chapter 3. K-finite eigenvectors in the Schrödinger model L2(C) -- 3.1. Result of this chapter -- 3.2. K Mmax-invariant subspaces Hl,k -- 3.3. Integral formula for the K Mmax-intertwiner -- 3.4. K-finite vectors fl,k in L2(C) -- 3.5. Proof of Theorem 3.1.1 -- Chapter 4. Radial part of the inversion -- 4.1. Result of this chapter -- 4.2. Proof of Theorem 4.1.1 (1) -- 4.3. Preliminary results on multiplier operators -- 4.4. Reduction to Fourier analysis -- 4.5. Kernel function Kl,k -- 4.6. Proof of Theorem 4.1.1 (2) -- Chapter 5. Main theorem -- 5.1. Result of this chapter -- 5.2. Radon transform for the isotropic cone C -- 5.3. Spectra of K'-invariant operators on Sp-2Sq-2 -- 5.4. Proof of Theorem 5.1.1 -- 5.5. Proof of Lemma 5.4.2 (Hermitian case q=2) -- 5.6. Proof of Lemma 5.4.2 (p,q&gt -- 2) -- Chapter 6. Bessel distributions -- 6.1. Meijer's G-distributions -- 6.2. Integral expression of Bessel distributions -- 6.3. Differential equations for Bessel distributions.

Chapter 7. Appendix: special functions -- 7.1. Riesz distribution x+ -- 7.2. Bessel functions J, I, K, Y -- 7.3. Associated Legendre functions P -- 7.4. Gegenbauer polynomials Cl -- 7.5. Spherical harmonics Hj(Rm) and branching laws -- 7.6. Meijer's G-functions Gp,qm,n(to1.5. x |to1.5. a1, @let@token , ap b1, @let@token , bq)to1.5. -- 7.7. Appell's hypergeometric functions F1, F2, F3, F4 -- 7.8. Hankel transform with trigonometric parameters -- 7.9. Fractional integral of two variables -- Bibliography -- List of Symbols -- Index.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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