Partial Fourier series on compact Lie groups

In this note we investigate the partial Fourier series on a product of two compact Lie groups. We give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. As applications, we will study conditions for the global solvability of an evolution equation defined on ${\mathbb{T}}^1\times {\mathbb{S}}^3$ and we will show that some properties of this evolution equation can be obtained from a constant coefficient equation.