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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 \(p>2\).

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
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