Question

Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. Let $G_\mathbb{Q}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

1: Let $G_\mathbb{Q}$ act on $B_p$ in the natural way. Then what is the structure of the stabilizer, $H_v$ of $v$, for $v|p$ an element of $B_v$?

Infinite primes are just embeddings of $\overline{\mathbb{Q}}$ into $\mathbb{C}$, so the case $p=\infty$ is really asking for the stabilizer of the action of left multiplication in $G_\mathbb{Q}$, and for finite $p$ it should be the inverse limit--over finite, Galois extensions of $\mathbb{Q}$--of the decomposition groups of primes $\mathfrak{p}|p$. However, actually getting ones hands on the structure of the stabilizers in question seems to be a hard question, at least no one I've talked to so far seems to know much more than the obvious things stated here.

2: Is is perhaps easier to describe the cosets $G/H_v$? If so, what should they look like?

Answer 1

If $p$ is a rational prime, then choosing a prime $v$ of $\overline{\mathbf{Q}}$ lying over $p$ amounts to choosing an embedding $i:\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}}_p$. This gives rise to a map $\varphi:G_{\mathbf{Q}_p}\rightarrow G_{\mathbf{Q}}$defined as follows: given $s$ in the source, $\varphi(s)$ is the unique automorphism of $\overline{\mathbf{Q}}$ such that $i\circ\varphi(s)=s\circ i$. This is continuous. Its image is the decomposition group (i.e. the stabilizer of) $v$, $G_v\subseteq G_\mathbf{Q}$. The kernel is the Galois group of $\overline{\mathbf{Q}}_p$ over $\mathbf{Q}_pi(\overline{\mathbf{Q}})$, which is trivial by Krasner's lemma. So you have an isomorphism $G_{\mathbf{Q}_p}\cong G_v$ (it is a homeomorphism because it is bijective with compact source and Hausdorff target).

The case of Archimedean primes is identical. In particular, $G_v$ for $v$ Archimedean has order $2$. When people talk about "choosing a complex conjugation" in $G_{\mathbf{Q}}$, they are referring to the choice of an embedding $\overline{\mathbf{Q}}\rightarrow\mathbf{C}$ which gives rise to an injection $\mathrm{Gal}(\mathbf{C}/\mathbf{R})\hookrightarrow G_{\mathbf{Q}}$, and the image of the unique non-trivial element of the source is the ``complex conjugation."

Whenever you have a, say, $\ell$-adic, Galois representation $\rho:G_\mathbf{Q}\rightarrow\mathrm{GL}_d(\mathbf{Q}_\ell)$, and a theorem talks about the local structure at $p$ of $\rho$, it means the restriction of $\rho$ to a decomposition group for a prime above $p$, which, by the paragraph above, can be identified with $G_{\mathbf{Q}_p}$. So a representation of $G_\mathbf{Q}$ gives rise to representations of $G_{\mathbf{Q}_p}$ for all $p$ by restriction...at least after choosing a decomposition group, which is unique up to conjugacy.

Answer 2

I do not understand what is $B_v$ in your notation, neither I see what you mean by "stabilizer of left multiplication".

Anyhow you are right in saying that $H_v$ is the inverse limit of decomposition groups and is isomorphic to the absolute Galois group of $\mathbb{Q}_p$. If $p=\infty$ you get a cyclic group with two elements, and if $p\neq \infty$ the structure of this Galois group is relatively well understood, although no explicit description is available. But since the structure of Galois groups of local fields is well-known, you see there is a sequence $$ 1\longrightarrow I_p\longrightarrow H_v\longrightarrow\widehat{\mathbb{Z}}\longrightarrow 1 $$ where $I_p$ is the inertia subgroup at $p$ (and not "at $v$" in the sense that they are all non-canonically isomorphic, the isomorphism being induced from the identification of your $H_v$ with the absolute Galois group of $\mathbb{Q}_p$ a choice of which is equivalent to having fixed your representative $v$). This inertia subgroup is a semi-direct product of the wild inertia subgroup $T_p$ (a pro-$p$ group which is difficult to put your hands on) and a tame inertia subgroup which is actually pro-cyclic of super-order prime to $p$.

Moreover, by local class field theory, you know that the abelianization of $H_v$ is isomorphic to $\widehat{\mathbb{Z}}^\times$. All this can be found in Serre's Corps Locaux, although only small hints are given at the infinite theory, or in Neukirch's Algebraische Zahlentheorie (I guess).