Let $G$ be a real reductive Lie group (e.g. $G=\operatorname{GL}(n,\mathbb{R})$). Let $\rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^\infty\subset V$ be the subspace of smooth vectors equipped with the Garding topology. Let $\rho^\infty$ be the natural representation of $G$ in $V^\infty$.

**Under what precise technical conditions the representations $\rho$ and $\rho^\infty$ have the same length?**

As far as I understand this situation is rather typical in the theory.

A reference would be very helpful.