Let $G = \{\rho \in \mathrm{Aut}(\mathbb{C}[[t]]) \,|\, \rho(t) \in t + t^2 \mathbb{C}[[t]] \}$ be a subgroup of continuos $\mathbb{C}-$automorphisms of $\mathbb{C}[[t]]$ and $\mathfrak{g} = t^2 \mathbb{C}[[t]] \, \partial_t \subseteq \mathrm{Der}(\mathbb{C}[[t]])$. Define $\exp: \mathfrak{g} \to G$ by the usual exponencial series $$ \exp(v(t) \partial_t) = \sum_{n \geq 0} \frac{1}{n!} (v(t) \partial_t)^n $$ This map is well-defined bijection. Let $r$ be a locally nilpotent representation of $\mathfrak{g}$, i.e., $r: \mathfrak{g} \to \mathfrak{gl}(V)$ such that for all $X \in \mathfrak{g}$ and $a \in V$ we have $r(X)^n a = 0$, $n\gg0$. I want to define a representation $R: G \to \mathrm{GL}(V)$ exponentiation $r$, i.e., $R(e^X) = e^{r(X)}$. As $\exp: \mathfrak{g} \to G$ is a bijetion and $r$ is locally nilpotent, we can define such $R$.
Question 1:: I want to prove that $R$ is a group homomorphism.
Let $w(t) \partial_t \in \mathfrak{g}$ s.t. $e^{w(t) \partial_t} = e^{u(t) \partial_t} e^{v(t) \partial_t}$. It is sufficient to prove that $w(t)$ is in the Lie subalgebra generated by $u(t)$ and $v(t)$. I would like to prove it without BCH formula. I could find the coefficients of $w(t)$ in terms of $u(t)$ and $v(t)$, but it's not an easy formula.
Question 2:: Are those all representations of $G$ ?
If $V$ is finite dimensional (over $\mathbb{C}$) I guess yes. Take $r(X) = \left.\frac{d}{dt}\right|_{t=0} R(e^{tX}) \in \mathfrak{gl}(V)$ (if $V$ is finite dimensional this derivative make sense), it is easy to see that $R(e^{X}) = e^{r(X)}$.
If $V$ is the limit of it's finite dimensional subrepresentation I think it will work too.