Compositio Mathematica

Representations modulo p of the p-adic group GL(2, F)

Marie-France Vignéras a1
a1 Université de Paris 7 – Denis Diderot, Institut de Mathématiques de Jussieu, 175/179 rue du Chevaleret, Paris 75013, France

Article author query
vigneras m-f   [Google Scholar] 


Let p be a prime number and let F be a local field with finite residual field of characteristic p. The Langlands local correspondence modulo $\ell \neq p $ for GL(n, F) is known for all integers $n\geq 1$ but the case $\ell=p$ is still mysterious even when n = 2 (the case n = 1 is given by the local class field theory). Any irreducible $\overline{\bf F}_p$-representation of GL(n, F) has a non-zero vector invariant by the pro-p-Iwahori subgroup I(1) and the pro-p-Iwahori–Hecke $\overline{\bf F}_p$-algebra ${\cal H}_{ \overline{\bf F}_p}(GL(n,F),I(1))$ plays a fundamental role in the theory of $\overline{\bf F}_p$-representations of G. We get when n = 2: (i) A bijection between the irreducible $\overline{\bf F}_p$-representations of dimension 2 of the Weil group $W(\overline F/ F)$ and the simple supersingular modules of the pro-p-Iwahori–Hecke $\overline{\bf F}_p$-algebra ${\cal H}_{\overline{\bf F}_p}(GL(2,F),I(1))$. (ii) A new proof of the Barthel–Livne classification of the irreducible non-supersingular $\overline{\bf F}_p$-representations of GL(2, F) using the I(1)-invariant functor. (iii) A bijection between the irreducible $\overline{\bf F}_p$-representations of GL(2, Qp) and the simple right ${\cal H}_{\overline{\bf F}_p}(GL(2,{\bf Q}_p),I(1))$-modules given by the I(1)-invariant functor, using the recent results of Breuil.

(Received September 20 2001)
(Accepted February 25 2002)

Key Words: mod p representations; p-adic group; pro-p-Iwahori–Hecke algebra.

Maths Classification

11S37; 11F70; 20C08; 20G05; 22E50.