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will be smooth, and we can define $(Xf)(g_\infty,g_f)$ to be the derivative of this function at $t=0$. Is this the correct way to get the $\mathfrak{g}$-action and hence the $(\mathfrak{g},K)$-module structure on the subspace of $K$-finite vectors?

will be smooth, and we can define $(Xf)(g_\infty,g_f)$ to be the derivative of this function at $t=0$.   Is this the correct way to get the $\mathfrak{g}$-action and hence the $(\mathfrak{g},K)$-module structure on the subspace of $K$-finite vectors? If so, is there a less ad hoc way to get the $\mathfrak{g}$-action? The fact this isn't an actual vector-valued derivative, like what you'd have with a smooth vector in e.g. a Banach space representation of $G_\infty$, is somehow unsatisfying to me, but perhaps I shouldn't expect anything like that since the point of view here does not involve a locally convex topology on the space $C^\infty(G(\mathbf{A}))$...of which I'm aware.

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adéles of $F$, $\mathbf{A}_f$ the finite adéles of $F$. Fix a maximal compact subgroup $K$ of $G_\infty$. We have Hecke algebras $H_\infty$ of $G_\infty$ (relative to $K$) and $H_f$ of $G(\mathbf{A}_f)$. The former is the one used by Flath and Borel-Jacquet in their Corvallis Volume 1 articles (not the one from Jacquet-Langlands) and is described in great detail in Chapter 1 of Knapp-Vogan's giant book on cohomological induction. It is the convolution algebra of left $K$-finite distributions on $G_\infty$ with support in $K$ (see Appendix B of Knapp-Vogan for distributions on manifolds). The algebra $H_f$ is the convolution algebra of $\mathbf{C}$-valued, locally constant, compactly supported functions on $G(\mathbf{A}_f)$. The important facts about these $\mathbf{C}$-algebras are that they have an "approximate identity" (a certain collection of idempotents) and a resulting notion of "smooth module" (see Chapter 1 of Knapp-Vogan for the definitions, although they use the term "approximately unital module"). One can prove that

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact subgroup $K$ of $G_\infty$. We have Hecke algebras $H_\infty$ of $G_\infty$ (relative to $K$) and $H_f$ of $G(\mathbf{A}_f)$. The former is the one used by Flath and Borel-Jacquet in their Corvallis Volume 1 articles (not the one from Jacquet-Langlands) and is described in great detail in Chapter 1 of Knapp-Vogan's giant book on cohomological induction. It is the convolution algebra of left $K$-finite distributions on $G_\infty$ with support in $K$ (see Appendix B of Knapp-Vogan for distributions on manifolds). The algebra $H_f$ is the convolution algebra of $\mathbf{C}$-valued, locally constant, compactly supported functions on $G(\mathbf{A}_f)$. The important facts about these $\mathbf{C}$-algebras are that they have an "approximate identity" (a certain collection of idempotents) and a resulting notion of "smooth module" (see Chapter 1 of Knapp-Vogan for the definitions, although they use the term "approximately unital module"). One can prove that