Pro-p-Iwahori invariants for SL_2 and L-packets of Hecke modules

TL;DR

This paper investigates the relationship between pro-p-Iwahori-Hecke algebras of SL_n(F) and GL_n(F), establishing an equivalence of categories and a correspondence linking Hecke modules to Galois representations.

Contribution

It introduces a categorical equivalence for SL_2(Q_p) Hecke modules and a numerical correspondence connecting supersingular Hecke modules with Galois representations.

Findings

Equivalence of categories between Hecke modules and smooth representations for SL_2(Q_p).

Numerical correspondence between supersingular Hecke modules and Galois representations.

Insights into the structure of pro-p-Iwahori-Hecke algebras and their modules.

Abstract

Let p be a prime number, and F a nonarchimedean local field of residual characteristic p. We explore the interaction between the pro-p-Iwahori-Hecke algebras of the group GL_n(F) and its derived subgroup SL_n(F). Using the interplay between these two algebras, we deduce two main results. The first is an equivalence of categories between Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SL_2(Q_p) and smooth mod-p representations of SL_2(Q_p) generated by their pro-p-Iwahori-invariants. The second is a "numerical correspondence" between packets of supersingular Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SL_n(F), and irreducible, n-dimensional projective Galois representations.