Question

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 $$ G^{ab} = G / [G, G] = \prod\limits_{p} GL_1( \mathbb{Z}_p).$$

How can we relate the groups $GL_n( \mathbb{Z}_p)$ to $G_p$ or $G$?

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 $GL_n(\mathbb{Z}_p)$?

Answer 1

This question has been solved by Paskunas in his PhD thesis:

http://front.math.ucdavis.edu/0306.5124

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.