Restriction problem for mod p representations of GL 2 over a finite field

Abstract Let F q ${\mathbb {F}_q}$ double struck upper F Subscript q be the finite field with q = p f $q = p^f$ q equals p Superscript f elements. We study the restriction of two classes of mod p representations of G q = GL 2 ( F q ) ${G_q} = \text {GL}_2({{\mathbb {F}_q}})$ upper G Subscript q Baseline equals GL Subscript 2 Baseline left parenthesis double struck upper F Subscript q Baseline right parenthesis to G p = GL 2 ( F p ) ${G_p} = \text {GL}_2({\mathbb {F}_p})$ upper G Subscript p Baseline equals GL Subscript 2 Baseline left parenthesis double struck upper F Subscript p Baseline right parenthesis . We first study the restrictions of principal series which are obtained by induction from a Borel subgroup B q ${B_q}$ upper B Subscript q . We then analyze the restrictions of inductions from an anisotropic torus T q ${T_q}$ upper T Subscript q which are related to cuspidal representations. Complete decompositions are given in both cases according to the parity of f