Deformation rings and parabolic induction

TL;DR

This paper establishes an isomorphism between deformation rings of supersingular representations and their parabolic inductions in p-adic groups, showing a unique lifting property for Banach representations.

Contribution

It proves that parabolic induction induces an isomorphism between deformation rings under certain conditions, revealing a unique lifting correspondence for Banach representations.

Findings

Parabolic induction defines an isomorphism between deformation rings.

Every Banach lift of an induced representation is uniquely determined by a lift of the original.

The result applies under mild genericity conditions.

Abstract

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 $\bar{\sigma}$.