Harmonic Analysis on Classical Groups

iO The simplest example of compact groups is {e }, on which there exists an orthonormal system inO { e }, n = 0, +/- 1, +/- 2 , ... , namely 1 J2" ." iO An immediate extension for {e } is group U", which consists of all n X n square matrices U satisfying ufj' = I, where fj' denotes the conjugate transpose matrix of U.