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 center $$ G^{ab} = \prod\limits_{p} GL_1( \mathbb{Z}_p).$$

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

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

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 center $$ G^{ab} = \prod\limits_{p} GL_1( \mathbb{Z}_p).$$

How can we relate the groups $GL_n( \mathbb{Z}_p)$ to $G_p$ or $G$? Which representations of $GL_n( \mathbb{Z}_p)$ occur as Galois representations? What representations of $GL_n( \mathbb{Z}/p)$ occur as Galois representations?

The motivation 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 actually induced by the maximal compact subgroup $GL_n(\mathbb{Z}_p)$.

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 center $$ G^{ab} = \prod\limits_{p} GL_1( \mathbb{Z}_p).$$

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