Suppose we are considering some pseudo-Riemannian manifold (spacetime) that has an underlying topology $SU(2)\times U(1)$ (which incidentally corresponds to a closed cyclic universe). Such a space is of course endowed with a metric tensor $g_{\mu\nu}$. Then we should be able to represent the metric tensor of a general space satisfying this via the generalized Fourier series (Via the Peter-Weyl theorem I believe):
$$g_{\mu\nu}=\sum_{m}^{\infty}A(m)Y_{\mu\nu}^{m}(x^{i})$$
Where the series indices m will in actuality be summed over multiple indices (four I believe n,m,l,k for example all with conditions between them? it's not important for this question). How do the symmetries underlying the manifold (the initial Lie group) manifest in the terms of the series? Via the Peter-Weyl theorem I'm led to believe there's a connection of some kind. Note: I attempted to ask this in a wholly different manner earlier but failed I believe in expressing the meat of my question. Also related: What does it mean for something to be $L^2(G)$ for compact G