We study deformations of smooth mod $p$ representations (and their duals) of a $p$-adic reductive group $G$. Under some mild genericity condition, we prove that parabolic induction with respect to a parabolic subgroup $P=LN$ defines an isomorphism between the universal deformation rings of a supersingular representation $\bar{ \sigma}$ of $L$ and of its parabolic induction $\bar{ \pi}$. As a consequence, we show that every Banach lift of $\bar{ \pi}$ is induced from a unique Banach lift of $\bar{ \sigma}$.
Categories: math.RT math.NT