On the restriction of some irreducible mod $p$ representations

TL;DR

This paper investigates how certain irreducible mod p representations of GL_2 over finite fields restrict to subgroups, providing explicit filtrations and characterizations for specific cases like q=4.

Contribution

It offers new explicit descriptions of the socle filtrations of restricted representations for q=p^2 and p^3, and identifies distinguished representations via characters.

Findings

Explicit socle filtration for q=4

Restriction behavior for q=p^2 and p^3

Identification of distinguished representations

Abstract

For a prime $p,$ let $\mathbb{F}_q$ be a finite extension of $\mathbb{F}_p.$ The restriction of an irreducible mod $p$ representation of $\text{GL}_2(\mathbb{F}_q)$ to its subgroup $\text{GL}_2(\mathbb{F}_p)$ can be seen as a tensor product of irreducible representations of $\text{GL}_2(\mathbb{F}_p).$ In this paper, we study the restriction of some of these representations of $\text{GL}_2(\mathbb{F}_q)$ to $\text{GL}_2(\mathbb{F}_p),$ for $q=p^2$ and $p^3$ using elementary tools and give explicit socle filtration when $q=4.$ We prove that when $q=p^2,$ a special class of representations of $\text{GL}_2(\mathbb{F}_q)$ are distinguished by suitable characters of $\text{GL}_2(\mathbb{F}_p).$