Perhaps this should be a comment, but my feeling is that your confusion comes from switching between equivalent notions. At the point where B-J introduce automorphic forms, there is no word about adeles. They make the connection to adeles in Section 4, in particular they say there "we want now to relate these automorphic forms to automorphic forms on $G_\infty$".

The original definition (that your refer to) is a traditional one using Hecke operators (in the form of convolving by finite measures on $K$) and derivatives (in the form of acting by $\mathfrak{g}$ and hence also by $U(\mathfrak{g})$). Smoothness ensures that the convolutions participating in the definition make sense, and then the definition is what it is. There is no need for any trickier auxiliary $\mathbf{C}$-vector space. If you wish, the "initial space" consists of $\mathbf{C}$-valued smooth functions on $G(\mathbf{R})$.

I also don't understand why you call the $\mathfrak{g}$-action (that you spell out) ad hoc. It is not ad hoc at all, it is the one induced by the right $G(\mathbf{R})$-action on the above mentioned "initial space". The action of $X\in\mathfrak{g}$ is simply differentiating a function on $G(\mathbf{R})$ in the direction of $X$.

The definition on the top of page 195 in B-J's article (in Corvallis 1) is ok, because $f\ast\xi$ is well-defined for any smooth $f:G(\mathbf{A})\to\mathbf{C}$ and any $\xi\in H$.

It suffices to verify the claim for pure tensors $\xi=\xi_\infty\otimes\xi_f$, in which case $f\ast\xi$ is $f\ast\xi_\infty$ convolved with $\xi_f\in H_f$ in the usual sense. The function $f\ast\xi_\infty$ exists by the smoothness assumption, while the integral defining the convolution converges, because $\xi_f:G(\mathbf{A}_f)\to\mathbf{C}$ is compactly supported and locally constant by definition.

Perhaps this should be a comment, but my feeling is that your confusion comes from switching between equivalent notions. At the point where B-J introduce automorphic forms, there is no word about adeles. They make the connection to adeles in Section 4, in particular they say there "we want now to relate these automorphic forms to automorphic forms on $G_\infty$".

The original definition (that your refer to) is a traditional one using Hecke operators (in the form of convolving by finite measures on $K$) and derivatives (in the form of acting by $\mathfrak{g}$ and hence also by $U(\mathfrak{g})$). Smoothness ensures that the convolutions participating in the definition make sense, and then the definition is what it is. There is no need for any trickier auxiliary $\mathbf{C}$-vector space. If you wish, the "initial space" consists of $\mathbf{C}$-valued smooth functions on $G(\mathbf{R})$.

I also don't understand why you call the $\mathfrak{g}$-action (that you spell out) ad hoc. It is not ad hoc at all, it is the one induced by the right $G(\mathbf{R})$-action on the above mentioned "initial space".

Perhaps this should be a comment, but my feeling is that your confusion comes from switching between equivalent notions. At the point where B-J introduce automorphic forms, there is no word about adeles. They make the connection to adeles in Section 4, in particular they say there "we want now to relate these automorphic forms to automorphic forms on $G_\infty$".

The original definition (that your refer to) is a traditional one using Hecke operators (in the form of convolving by finite measures on $K$) and derivatives (in the form of acting by $\mathfrak{g}$ and hence also by $U(\mathfrak{g})$). Smoothness ensures that the convolutions participating in the definition make sense, and then the definition is what it is. There is no need for any trickier auxiliary $\mathbf{C}$-vector space. If you wish, the "initial space" consists of $\mathbf{C}$-valued smooth functions on $G(\mathbf{R})$.

I also don't understand why you call the $\mathfrak{g}$-action (that you spell out) ad hoc. It is not ad hoc at all, it is the one induced by the right $G(\mathbf{R})$-action on the above mentioned "initial space". The action of $X\in\mathfrak{g}$ is simply differentiating a function on $G(\mathbf{R})$ in the direction of $X$.