Pro-p Iwahori Invariants

Let $F$ be the field of $p$-adic numbers (or, more generally, a non-archimedean local field) and let $G$ be $\mathrm{GL}_n(F)$ (or, more generally, the group of $F$-points of a split connected reductive group). In the framework of the local Langlands program, one is interested in studying certain classes of representations of $G$ (and hopefully in trying to match them with certain classes of representations of local Galois groups).

In this talk, we are going to focus on the category of smooth representations of $G$ over a field $k$. An important tool to investigate this category is given by the functor that, to each smooth representation $V$, attaches its subspace of invariant vectors $V^I$ with respect to a fixed compact open subgroup $I$ of $G$. The output of this functor is actually not just a $k$-vector space, but a module over a certain Hecke algebra. The question we are going to attempt to answer is: how much information does this functor preserve or, in other words, how far is it from being an equivalence of categories? We are going to focus, in particular, on the case that the characteristic of $k$ is equal to the residue characteristic of $F$ and $I$ is a specific subgroup called "pro-$p$ Iwahori subgroup".