Deformation rings and parabolic induction

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 $\bar{\sigma }$.

arXiv comment28 pages, minor changes, final version

References in corpus (7)

• The classification of irreducible admissible mod p representations of a p-adic GL_n
• A classification of irreducible admissible mod p representations of p-adic reductive groups
• Parabolic induction and extensions
• Extensions for supersingular representations of $G{L}_2(Q_p)$
• Functorial properties of generalised Steinberg representations
• Modulo $p$ representations of reductive $p$-adic groups: functorial properties
• On the exactness of ordinary parts over a local field of characteristic $p$

Cited by in corpus (1)

• From $p$-modular to $p$-adic Langlands correspondences for $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$: deformations in the non-supercuspidal case