Let $\pi$ is an automorphic representation of reductive group $G$.
Then we can decompose $\pi = \otimes \pi_v$ as restricted tensor product by Flath theorem.
I am wondering whether $\text{Hom}_G(\pi,\mathbb{C}) \simeq \otimes \text{Hom}_{G_v}(\pi_v,\mathbb{C})$? If is not, which one is bigger among them?
