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.

qualitative(not formulaic) change-of-variables (a.k.a., inverse function theorem...) should suffice. It's not about Lie groups, etc. Asking or wanting a formula is a bit misguided, even if natural. – paul garrett Jun 10 '13 at 23:56