Restriction problem for mod $p$ representations of $\text{GL}_2$ over a finite field

TL;DR

This paper investigates how certain mod p representations of GL_2 over finite fields restrict to smaller subgroups, providing explicit decompositions based on the structure of the field extension.

Contribution

It offers a detailed analysis of the restriction of principal series and cuspidal representations of GL_2 over finite fields, with explicit decompositions depending on the extension degree.

Findings

Complete restriction decompositions for principal series representations.

Explicit orbit decompositions for different cases based on the parity of f.

Methodology using Mackey theory and orbit analysis in projective lines.

Abstract

Let $\mathbb{F}_q$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod $p$ representations of $G_q = \text{GL}_2({\mathbb{F}_q})$ to $G_p = \text{GL}_2(\mathbb{F}_p)$. We first study the restrictions of principal series which are obtained by induction from a Borel subgroup $B_q$. We then analyze the restrictions of inductions from an anisotropic torus $T_q$ which are related to cuspidal representations. Complete decompositions are given in both cases according to the parity of $f$. The proofs depend on writing down explicit orbit decompositions of $G_p \backslash G_q / H$ where $H = B_q$ or $T_q$ using the fact that $G_q / H$ is an explicit orbit in a certain projective line, along with Mackey theory.