Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups

Abstract

The Plancherel formula and the inversion formula for Weyl trans-forms on compact and Hausdorff groups are given. A formula expressing the relationships of the wavelet constant, the degree of the irreducible and unitary representation and the volume of an arbitrary compact and Hausdorff group is derived. The role of the Weyl transforms in the derivation of the formulas for the heat kernels of Laplacians on compact Lie groups is explicated. The Green functions and the Riemann zeta functions of Laplacians on compact Lie groups are constructed using the corresponding heat kernels.