On the Fourier transform of a compact semisimple Lie group

Source abstract

Abstract We develop a concrete Fourier transform on a compact Lie group by means of a symbol calculus, or *-product, on each integral co-adjoint orbit. These *-products are constructed by means of a moment map defined for each irreducible representation. We derive integral formulae for these algebra structures and discuss the relationship between two naturally occurring inner products on them. A global Kirillov-type character is obtained for each irreducible representation. The case of SU (2) is treated in some detail, where some interesting connections with classical spherical trigonometry are obtained.