Suppose $\mathcal{G}_k$ is the absolute Galois group of a number field $k$.

$\mathcal{G}_k$ is a topological group, with profinite topology. How does the theory of harmonic analysis of regular representations of locally compact groups apply to it? Which function spaces on $\mathcal{G}_k$ is it meaningful to consider; how do (left or right) regular representations of $\mathcal{G}_k$ on them decompose into irreducibles; which irreducibles occur; and what is the analog of the Plancherel measure?