Deformation rings and parabolic induction

Hauseux, Julien; Schmidt, Tobias; Sorensen, Claus

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 2, pp. 695-727.

Nous étudions les déformations des représentations lisses modulo $p$ (et leurs duaux) d’un groupe réductif $p$-adique $G$. Sous une hypothèse de généricité faible, nous prouvons que le foncteur d’induction parabolique relatif à un sous-groupe parabolique $P=LN$ induit un isomorphisme entre l’anneau de déformation universel d’une représentation supersingulière $\overline{\sigma }$ de $L$ et de son induite parabolique $\overline{\pi }$. En conséquence, nous montrons que tout relèvement continu de $\overline{\pi }$ est induit à partir d’un unique relèvement continu de $\overline{\sigma }$.

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 $\overline{\sigma }$ of $L$ and of its parabolic induction $\overline{\pi }$. As a consequence, we show that every Banach lift of $\overline{\pi }$ is induced from a unique Banach lift of $\overline{\sigma }$.

Classification : 22E50, 11F70
Mots clés : $p$-adic reductive groups, smooth representations, $\protect \mathfrak{m}$-adically continuous representations, parabolic induction, deformations

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