Some notations:
- $F$ a global field with places $v$.
- $\chi_1$, $\chi_2$ two Hecke characters $A_F^\times/F^\times\to\mathbf{C}^\times$.
- $K$ the compact maximal subgroup $\prod_{v\text{ real}}O(2)\prod_{v\text{ complex}}U(2)\prod_{v<\infty}GL_2(\mathfrak{o}_v)$ of $GL_2(A_F)$.
- $V_{\chi_1,\chi_2}$ the $GL_2(A_F)$-representation
$$V_{\chi_1,\chi_2}=\{\text{smooth and $K$-finite }f:GL_2(A_F)\to\mathbf{C}\mid f(\begin{bmatrix} a&b\\ 0&d \end{bmatrix}g)=\chi_1(a)\chi_2(d)\left|\frac{a}{d}\right|_{A_F}^{1/2}f(g) \}.$$
In Bump's book p.350, $V_{\chi_1,\chi_2}$ is decomposed into the restricted tensor product of local factors $$ V_{\chi_1,\chi_2}=\bigotimes_v'V_{\chi_{1,v},\chi_{2,v}} $$ where $$V_{\chi_{1,v},\chi_{2,v}}=\{\text{smooth and $K_v$-finite }f:GL_2(F_v)\to\mathbf{C}\mid f(\begin{bmatrix} a&b\\ 0&d \end{bmatrix}g)=\chi_{1,v}(a)\chi_{2,v}(d)\left|\frac{a}{d}\right|_{v}^{1/2}f(g)\}.$$
and $\bigotimes_v'V_{\chi_{1,v},\chi_{2,v}}$ is generated by the pure tensors $\bigotimes f_v$, where $f_v$ is spherical for almost all $v$ so that $f_v(k)=1$ for $k$ in the compact maximal subgroup $K_v$ of $GL_2(F_v)$.
How to prove this decomposition? There is a natural map $$ \bigotimes_v'V_{\chi_{1,v},\chi_{2,v}}\to V_{\chi_1,\chi_2},\quad \bigotimes f_v\mapsto(g\mapsto\prod_v f_v(g_v)) $$
but the injectivity and surjectivity seem non-trivial.