I got stuck in an apparently trivial point within the proof of Lemma 3.13 on p. 55 of Knapp's Representation Theory of Semisimple Groups. The author concludes in the first paragraph that $f_v$ must be of class $\mathcal{C}^1$. In local coordinates around the identity element, this amounts to the existence and continuity of all directional derivatives. Explicitly, one has to show for each direction $X\in\mathfrak{g}$ that the map
$$Y\mapsto\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\Phi(e^{Y+tX})v$$
makes sense and is continuous in a neighborhood of $0$ in $\mathfrak{g}$. This would follow at once from the hypothesis if we had $e^{Y+tX}=e^Y e^{tX}$ which is of course false in general. Actually, I cannot even see at the moment why the maps above are meaningful.
Recently, I asked essentially the same question to some knowledgeable people around me and also here in MathOverflow, and got quite different answers. Some referred to the Campbell-Baker-Hausdorff formula while others made no mention of it. I pondered upon each answer, but unfortunately none of them led me to an understanding of the issue.