Shimura in his book Introduction to the theory of automorphic functions on p. 143, section 6.4, defines $$G_{\infty}=\operatorname{GL}_2(\mathbb R)$$ $$G_{\infty+}=\{x\in G_{\infty}: \det x >0\}$$ and $$G_p=\operatorname{GL}_2(\mathbb Q_p).$$ Further he defines $G_A$ as the group consisting of all elements $x=(\cdots,x_p,\cdots,x_\infty)$ of $$\prod_pG_p\times G_\infty$$ such that $x_p\in \mathbb Z_p$ for all but finite number of $p$.
Then he defines $G_0$ to be the archimedean part of $G_A$, that is, the set of all elements of $G_A$ whose $x_\infty$ component is $1$.
Finally he puts $$G_{A+}=G_0G_{\infty+}.$$
Thus last product does not make sense to me. In which group are we taking it?