# Pro-p Iwahori Invariants

Date: Thu, Mar 21, 2024

Location: PIMS, University of British Columbia, Online

Conference: UBC Number Theory Seminar

Subject: Mathematics, Number Theory

Class: Scientific

### Abstract:

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".