## 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:

 2.2e+51 Representations of Lie and linear algebraic groups over local fields

### Keywords:

mod $$p$$-representation; pro-$$p$$-Iwahori group
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### References:

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