Blocks for mod $p$ representations of $G{L}_2(Q_p)$

Abstract

Let ${\pi }_1$ and ${\pi }_2$ be absolutely irreducible smooth representations of $G=G{L}_2(Q_p)$ with a central character, defined over a finite field of characteristic $p$. We show that if there exists a non-split extension between ${\pi }_1$ and ${\pi }_2$ then they both appear as subquotients of the reduction modulo $p$ of a unit ball in a crystalline Banach space representation of $G$. The results of Berger-Breuil describe such reductions and allow us to organize the irreducible representation into blocks. The result is new for $p=2$, the proof, which works for all $p$, is new.

Added an appendix, minor changes

Cited by in corpus (6)

• The p-adic local Langlands correspondence for GL_2(Q_p)
• The image of Colmez's Montreal functor
• Finiteness properties of the category of mod $p$ representations of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$
• Higher $\mathcal{L}$-invariants for ${\mathrm{GL}}_3({\mathbb{Q}}_p)$ and local-global compatibility
• ${\mathrm{GL}}_2({\mathbb{Q}}_p)$-ordinary families and automorphy lifting
• On Extensions of supersingular representations of ${\mathrm{SL}}_2({\mathbb{Q}}_p)$