On the $I(1)$-invariants: Non-abelian Hecke algebra case

Abstract

Let $F$ be a finite extension of ${\mathbb{Q}}_p$. The so-called supersingular representations are the basic building blocks in the theory of mod $p$ representations of ${\mathrm{GL}}_2(F)$. The space of pro-$p$-Iwahori invariants of a universal module played a crucial role in the construction of the supersingular representations of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$. In this paper, we give an explicit description of the pro-$p$-Iwahori invariants of the universal module ${\pi }_r$ for $r=0,q-1$ using the Iwahori-Hecke model. We also determine the action of the pro-$p$-Iwahori-Hecke algebra on these newly found invariants. As an application, we recover ${\pi }_r$ functorially from its space of $I(1)$-invariants and extend a theorem of Ollivier for any totally ramified extension of ${\mathbb{Q}}_p$ other than itself.

27 pages, 1 figure, and 1 table