Consider the absolute Galois group $G = \mathrm{Gal}(\overline{\mathbb{Q}}: \mathbb{Q})$ and $G\_p = \mathrm{Gal}(\overline{\mathbb{Q}\_p}: \mathbb{Q}\_p)$.


Abelian class field theory gives us for the abelinisation
$$ G^{ab} = G / [G, G] = \prod\limits\_{p} GL\_1( \mathbb{Z}\_p).$$



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> How can we relate the groups $GL\_n( \mathbb{Z}\_p)$ to $G\_p$ or $G$?
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The motivation of my question in a nutshell: Langlands program relates automorphic representations to Galois representations. Automorphic representations factor in representations of $GL\_n( \mathbb{Q}\_p)$. Some representations (the cuspidal ones) are induced representations from the maximal compact subgroup $GL\_n(\mathbb{Z}\_p)$. It seems easier to relate them directly. Since the local Langlands conjecture have been proven for $GL\_n(\mathbb{Q}\_p)$ and the dual of $GL(n, \mathbb{Q}\_p)$ is described by the dual of $GL\_n(\mathbb{Z}\_p)$, what can we deduce about the relation between $G\_p$ and $GL\_n(\mathbb{Z}\_p)$?



What is the relation of the absolute Galois group and classical profinite groups?
This question has been solved by Paskunas in his PhD thesis, *Unicity of types for supercuspidals*, arXiv:[math/0306124](https://arxiv.org/abs/math/0306124).


Corollary 8.2 of this reference gives an "inertial Galois correspondence" between supercuspidal types of the form $({\rm GL}(n,{\mathbb Z}\_p), \lambda )$ and certain representations of the local inertial group.


The case of non-supercuspidal types is a work in progress by Shaun Stevens and William Conley, certainly to appear somewhere soon.


