Let $G$ be a reductive group over $\mathbb Q$, $\mathfrak g=Lie(G)$ be its Lie algebra, $\mathfrak g\otimes_{\mathbb R} \mathbb C$ be its complexification. A definition of automorphic representation on $G$ is that it is an irreducible admissible $(\mathfrak g_{\mathbb C},K)\times G(\mathbb A^{\infty})$ module that is isomorphic to a subquotient of the space of automorphic forms on $G$. (e.g. this definition is used in page 146 of Getz and Hahn's book)
Why do we consider $(\mathfrak g_{\mathbb C},K)\times G(\mathbb A^{\infty})$-modules, rather than $(\mathfrak g,K)\times G(\mathbb A^{\infty})$-modules?
