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

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.