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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)?

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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.

So it is a purely representation theoretic reason to introduce this terminology.

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