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