rt.representation theory – Under what conditions representations of reductive Lie group in Banach space and in its Garding space have the same length?

Let $G$ be a real reductive Lie group (e.g. $G=GL(n,mathbb{R})$). Let $rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^inftysubset V$ 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?

A reference would be very helpful.