Bijection between two orbit's spaces Take $\mathbb A_\mathbb Q$ the adele ring of $\mathbb Q$ and let
$$\vert \cdot\vert:\mathbb A_\mathbb Q\to \mathbb R_{>0}, x=(x_p)\mapsto \prod_{p\leq \infty}\vert x_p\vert_p$$ the absolute value induced by each $p$-adic absolute value.
Define $$GL_2(\mathbb A_\mathbb Q)^1=\{g\in GL_2(\mathbb A_\mathbb Q):\vert det(g)\vert=1\}.$$
Could you give me some idea to see that there exist a bijection between
$$GL_2(\mathbb Q)\setminus GL_2(\mathbb A_\mathbb Q)^1$$
and
$$GL_2(\mathbb Q)Z(GL_2(\mathbb R))\setminus GL_2(\mathbb A_\mathbb Q),$$
with $Z(GL_2(\mathbb R))$ the center of $GL_2(\mathbb R)$.
 A: I think the center $Z(\mathrm{GL}_{2}(\mathbb{R}))$ in your bijection should be replaced by its identity component $Z(\mathrm{GL}_{2}(\mathbb{R}))^{\circ}$.
Otherwise you will get the bijection:
$$\pm\mathrm{Id}_{2}\mathrm{GL}_{2}(\mathbb{Q})\backslash\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})^{1}\simeq\mathrm{GL}_{2}(\mathbb{Q})Z(\mathrm{GL}_{2}(\mathbb{R}))\backslash\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}}),$$
where $\pm\mathrm{Id}_{2}\in\mathrm{GL}_{2}(\mathbb{R}).$
Consider the composition $\varphi$ of the embedding $\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})^{1}\hookrightarrow \mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ with the quotient morphism $\mathrm{GL}_{2}(\mathbb{A_{\mathbb{Q}}})\rightarrow \mathrm{GL}_{2}(\mathbb{Q})Z(\mathrm{GL}_{2}(\mathbb{R}))^{\circ}\backslash\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}}).$
The kernel of $\varphi$ is $\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})^{1}\cap \mathbb{GL}_{2}(\mathbb{Q})Z(\mathrm{GL}_{2}(\mathbb{R}))^{\circ}$. Notice that $\mathrm{GL}_{2}(\mathbb{Q})$ is contained in $\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})^{1}$ naturally. An element $\mathrm{diag}(r,r),r>0$ in the center of $\mathrm{GL}_{2}(\mathbb{R})$ is in the kernel of $\varphi$ if and only if $r^{2}=1$, thus $\ker\varphi\cap Z(\mathrm{GL}_{2}(\mathbb{R}))^{\circ}=\mathrm{Id}_{2}$. So $\ker\varphi=\mathrm{GL}_{2}(\mathbb{Q}).$
On the other side, for any $X\in\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ its norm $|\det X|$ is a positive number.
Take $r=\sqrt{\det X}$ and then $\mathrm{diag}(r,r)^{-1}X\in\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})^{1}$, which implies that $\varphi$ is surjective and we obtain the bijection between these two quotient spaces.
