On a Theorem of Shintani

Let χ be an irreducible character of Gd = GLn( qd) invariant under the automorphism φ of Gd induced by the field automorphism qd → qd, x xq, and let e be a divisor of d. By a theorem of Shintani, there exists an extension e of χ to Gd φe whose Shintani descent to Ge is, up to a sign , an irreducible character of Ge. It is shown in this paper that e may always be chosen such that = 1. With this particular choice, e is the restriction of 1. Our methods rely on the work of Digne and Michel on Deligne–Lusztig theory for nonconnected reductive groups.