# What role do $(\mathfrak{g},K)$- modules play in the construction of automorphic vector bundles

Looking at the connection between modular forms as sections and automorphic representations it is to me somewhat clear why automorphic representations are (demanded to be) admissible $G(\mathbb{A}^\infty)$ modules.

Is there a similar reason for the $(\mathfrak{g},K)$-module structure(like some nice analytic structures of the total space corresponding to the $(\mathfrak{g},K)$-module) or does this part rather come from a purely representation theoretic side (eg. classification reasons)?

Given any admissible representation of $G(\mathbb{R})$ one can construct a $(\mathfrak{g}, K)$ module from it. Isomorphism of the admissible representations is not the same as isomorphism of the $(\mathfrak{g}, K)$ modules (called infinitesimal isomorphism), but these notions agree for unitary representations. Therefore we lose nothing in looking at $(\mathfrak{g}, K)$ modules if we're working with forms that are going to have unitary real representation.