Let $G$ be a real reductive Lie group, $P=MN$ its parabolic subgroup with Levi decomposition. Suppose $\sigma$ is a smooth admissible irreducible representation of $M$, extend this to $P$ by letting $N$ act trivially. Form the unitarily induced representation $Ind_P^G(\sigma)$.
My question is what is the contragredient representation (smooth admissible dual) of $Ind_P^G(\sigma)$ in terms of $\sigma$ ? In particular, is it equal to $Ind_P^G(\sigma')$, where $\sigma'$ is the smooth admissible dual of $\sigma$?