Gelfand-Kirillov dimension and mod p cohomology for GL2

TL;DR

This paper investigates the Gelfand-Kirillov dimension of certain smooth representations of GL2 over local fields, linked to mod p Galois representations, revealing that many have maximal dimension equal to the degree of the local field extension.

Contribution

It establishes that many admissible smooth representations associated to a given mod p Galois representation have maximal Gelfand-Kirillov dimension, connecting representation theory with Galois and cohomological data.

Findings

Many representations have Gelfand-Kirillov dimension equal to [F_v:Q]

Results relate mod p cohomology to representation dimensions

Provides new insights into the structure of mod p representations of GL2

Abstract

Let $p$ be a prime number, $F$ a totally real number field unramified at places above $p$ and $D$ a quaternion algebra of center $F$ split at places above $p$ and at no more than one infinite place. Let $v$ be a fixed place of $F$ above $p$ and $\overline{r} : {\rm Gal}(\overline F/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ an irreducible modular continuous Galois representation which, at the place $v$, is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of $\mathrm{GL}_2(F_v)$ over $\overline{\mathbb{F}}_p$ associated to $\overline{r}$ in the corresponding Hecke-eigenspaces of the mod $p$ cohomology have Gelfand--Kirillov dimension $[F_v:\mathbb{Q}]$, as well as several related results.