Let $G$ a linear Lie group with lie algebra $\mathfrak{g}$. If $(\pi, \mathcal{H})$ is a irreducible representation of $G$. Does the irreducibility of $\pi$ imply the irreducibility of derived representation $d\pi$?
$$d\pi:\mathfrak{g}\longrightarrow End(\mathcal{H}),\,\, X\longmapsto \frac{d}{dt}\Bigr|_{t=0}\pi(\exp(tX))$$
If all vectors in $\pi$ are differentiable, then yes. It suffices to show that a closed subspace $V \subset \mathcal{H}$ which is invariant under $G$ is also invariant under $\mathfrak{g}$. Let $v \in V$ and let $X \in \mathfrak{g}$. Then by $G$-invariance of $W$, we have $t^{-1}(\pi(\exp{tX})v-v) \in V$ for all $t \in \mathbb{R}^{\times}$. Since $V$ is assumed to by closed, the limit as $t \rightarrow 0$ also belongs to $V$.
If $G$ is conected and $\mathcal{H}$ is finite dimensional a (necessarily closed) subsapce $W$ of $\mathcal{H}$ which is $\mathfrak{g}$-invariant, is also $G$-invaraint.