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

Published online by Cambridge University Press:  04 December 2007

Marie-France Vignéras
Affiliation:
Université de Paris 7 – Denis Diderot, Institut de Mathématiques de Jussieu, 175/179 rue du Chevaleret, Paris 75013, Francevigneras@math.jussieu.fr
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Abstract

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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.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2004