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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?

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    $\begingroup$ The absolute Galois group (of any field) is not only locally compact, it is compact. This makes the harmonic analysis of it rather trivial. For instance, the regular representation is the Hilbert direct sum of each irreducible representation, which all have finite dimension, each of them with multiplicity equal to its dimension... $\endgroup$
    – Joël
    Aug 15 '20 at 13:25
  • $\begingroup$ And if it is true that any finite group is a quotient of the absolute galois group, then there is really no hope of saying anything useful about which irreps can appear. $\endgroup$
    – Asvin
    Aug 15 '20 at 20:56
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The absolute Galois group (of any field) is not only locally compact, it is compact. This makes the harmonic analysis of it completely solved by Peter-Well theory.

In particular, the regular representation is the Hilbert direct sum of each irreducible representation, which all have finite dimension, each of them with multiplicity equal to its dimension.

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