$\DeclareMathOperator\Ind{Ind}$Let $G$ be a reductive group over a $p$-adic local field $F$, and $P=MN$ a parabolic subgroup.

Let $\sigma$ be an irreducible representation of $M(F)$ and 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 $f \in \pi$ such that $f(e)=\nu$?

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