The functor of invariants under the action of the pro-\(p\)-Iwahori of \(\mathrm{GL}_2(F)\). (Le foncteur des invariants sous l’action du pro-\(p\)-Iwahori de \(\mathrm{GL}_{2}(F)\).) (French) Zbl 1181.22017

Let \(F\) be a non-archimedean local field with residual characteristic \(p\) and let \(G = \text{GL}_2(F)\). In the study of the representations of \(G\) over \(\overline {{\mathbb F}}_p\) an important role is played by the functor of invariants under the action of the pro-\(p\)-Sylow group \(I(1)\) of the Iwahori subgroup of \(G\). It is proved in this article that, for \(F = {\mathbb Q}_p\), the functor of \(I(1)\)-invariants gives an equivalence of categories between the smooth \(\overline {{\mathbb F}}_p\)-representations of \(G\) generated by their \(I(1)\)-invariants and the right modules over the Hecke algebra \({\mathcal H} = {\mathcal H}_{\overline{{\mathbb F}}_p}(G,I(1))\). Let \(M\) be a right \({\mathcal H}\)-module. One has to prove that the obvious homomorphism of \({\mathcal H}\)-modules from \(M\) to the space of \(I(1)\)-invariants of \(M\otimes_{{\mathcal H}} \overline{{\mathbb F}}_p[I(1)\backslash G]\) is an isomorphism. The proof uses the decomposition of \(\overline{{\mathbb F}}_p[I(1)\backslash G]\) as a left \(G\)-module and a filtration of it as a right \({\mathcal H}\)-module. It requires some calculations in the tree of \(\text{PGL}_2(F)\).
It is also shown that the functor considered here is not an equivalence of categories when \(F\) has residual field \({\mathbb F}_{p^m}\), \(m>1\), or when \(F = {\mathbb F}_p((T))\) with \(p>2\).


22E50 Representations of Lie and linear algebraic groups over local fields
Full Text: DOI


[1] DOI: 10.1215/S0012-7094-94-07508-X · Zbl 0826.22019
[2] Barthel L., J. Number Th. pp 55– (1995)
[3] DOI: 10.1023/A:1026191928449 · Zbl 1044.11041
[4] Ollivier R., Comp. Math. 143 (2) pp 703– (2007)
[5] Paskunas V., S.) pp 99– (2004)
[6] DOI: 10.1007/BF02699536 · Zbl 0892.22012
[7] Serre J.-P., Astérisque pp 46– (1977)
[8] DOI: 10.1112/S0010437X03000071 · Zbl 1049.22010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.