Peter-Weyl theorem can be interpreted as a generalization of Fourier analysis. But, usually, this theorem is applied to functions on some compact group G.
I want to know that we can generalize this theorem to for spinor, gauge field,... if the compact group G admit a spin structure.
For example, for the sphere, we know spin-weighted spherical harmonic functions. If we apply the Peter-Weyl theorem to the sphere, we can obtain the usual spherical harmonic functions. Can we understand the spin-weighted spherical harmonic functions from the (generalized) Peter-Weyl theorem?