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Special Functions and Linear Representations of Lie Groups
Jean Dieudonné
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1980; 59 pp; softcover
Number: 42
Reprint/Revision History:
reprinted 1982
ISBN-10: 0-8218-1692-6
ISBN-13: 978-0-8218-1692-9
List Price: US$18
Member Price: US$14.40
All Individuals: US$14.40
Order Code: CBMS/42
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Reviews

"Presented in a clear, self-contained, and well-organized way, these introductory level lectures are highly recommended for individual or in-class use for acquiring basic knowledge of the groups \({\mathbf SU}(2)\), \({\mathbf SO}(3)\), \({\mathbf SL}(2, \mathbf R)\), and their function theories. Covering representations of \({\mathbf SU}(2)\), spherical harmonics on \(S_n\), representations of the group \({\mathbf SL}(2, \mathbf R)\) and spherical functions with respect to its maximal compact subgroup \({\mathbf SO}(2)\), these lectures also take account of some well known functions exhibited as spherical or matrix-entry functions of unitary representations--for example, Jacobi polynomials and Legendre functions appear in this way for \({\mathbf SU}(2)\)."

-- C. F. Dunkl, Mathematical Reviews

Table of Contents

  • Introduction
  • Representations of \(\mathbf {SU}(2)\)
  • The general theory of linear representations of compact groups
  • Lie theory of representations of compact connected Lie groups
  • Induced representations of compact groups
  • Spherical functions on compact groups
  • Examples; spherical harmonics
  • The general theory of spherical functions
  • Fourier and Plancherel transforms
  • Extension of the Plancherel transform
  • The subtleties of harmonic analysis
  • Differential properties of spherical functions on Lie groups
  • Spherical functions on semisimple Lie groups
  • More on \(\mathbf {SL}(2, \mathbf R)\)
  • Automorphic functions
  • Groups of isometries and Bessel functions
  • Other special functions
  • References
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