In this question, we will follow the notations and definitions from the book "Automorphic Representations and L-Functions for the General Linear Group" by Goldfeld and Hundley. For simplicity, let's focus of $GL_2$. The definition of an (cuspidal) automorphic representation with central character $\omega$ is given in Definition 5.1.12, (5.1.14), where $\omega$ is a unitary Hecke character $$\omega: \mathbb{Q}^{\times} \backslash \mathbb{A}_{\mathbb{Q}}^{\times} \rightarrow \mathbb{C}^{\times}$$

i.e. it is a smooth $(\mathfrak{g},K_{\infty}) \times GL(2,\mathbb{A}_{\text{finite}})$-module which is also isomorphic to a subquotient of the complex vector space of adelic (cusp) automorphic forms. As I literally know nothing about the theory of automorphic representations, I have two questions to bother the community.

Suppose we have two cuspidal automorphic representations with the same central character $\omega$, and if they are isomorphic as $(\mathfrak{g},K_{\infty}) $ modules, are they isomorphic as automorphic representations? If not, do the two automorphic representations have any relations?

Given a $(\mathfrak{g},K_{\infty}) $ module, how to check whether there exists a cuspidal automorphic representation whose local representation at $\infty$ is just this module? E.g. in order for this to be true, are there restrictions on this module? On the other hand, if there exists such a cuspidal automorphic representation, is it unique?

Any references are welcomed!