Let $G$ be a linear algebraic group over $\mathbb C$. Let $\psi$ be a finite dimensional regular representation of $G$ into $GL(V)$.
Suppose $G$ is connected. I would like to show for $v$ in $V$ the following are equivalent:
1) $\psi(g)(v)=v$ for all $g$ in $G$.
2) $d\psi(X)(v)=0$ for all $X$ in the Lie algebra of $G$.
For 1) implies 2), can I just say that the representation is "constant" for all $g$ and so the derivative is 0? Is that the right intuition? Is there a more formal way to show it? Where does connectedness come in?
For the other way I dont see how to use any of the theorems I know to take the Lie algebra information and bring it to the group. My intuition is that, because we are connected, and "locally constant", we are constant everywhere.