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.