Irreducible modular representations of the Borel subgroup of GL_2(Q_p)

TL;DR

This paper demonstrates that all infinite dimensional smooth irreducible representations of the Borel subgroup of GL_2(Qp) with a central character can be constructed from irreducible Galois or Weil group representations, extending Colmez's work.

Contribution

It proves that every such Borel subgroup representation arises from Galois or Weil group representations, generalizing Colmez's construction to algebraically closed fields of characteristic p.

Findings

All irreducible B_2(Qp) representations with central character are obtained from Galois or Weil group representations.

The result extends Colmez's correspondence to algebraically closed fields of characteristic p.

The proof unifies the construction of these representations via Fontaine's (phi,Gamma)-modules.

Abstract

Let E be a finite extension of Fp. Using Fontaine's theory of (phi,Gamma)-modules, Colmez has shown how to attach to any irreducible E-linear representation of Gal(Qpbar/Qp) an infinite dimensional smooth irreducible E-linear representation of B_2(Qp) that has a central character. We prove that every such representation of B_2(Qp) arises in this way. Our proof extends to algebraically closed fields E of characteristic p. In this case, infinite dimensional smooth irreducible E-linear representations of B_2(Qp) having a central character arise in a similar way from irreducible E-linear representations of the Weil group of Qp.