Let $G$ be a reductive group over a $p$-adic localfield $F$ and $P=MN$ its parabolic subgroup.

Let $\sigma$ be a irreducible representation of $M(F)$ and we consider its unnormalized induced representation $Ind_P^G(\sigma)$. Let $\pi$ be a subrepresentation of $Ind_P^G(\sigma)$. For an arbitrary element $\nu$ in $\sigma$, can we choose a $f \in \pi$ such that $f(e)=\nu$?

Some paper says that it is possible because $\sigma$ is irreducible and by using right translation of $M$. But I don't know why it holds for sub representation $\pi$ instead of $Ind_P^G(\sigma)$.

Any comments will be appreciated!