Restriction problem for mod <i>p</i> representations of 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.