Shintani lifting and real-valued characters

We study Shintani lifting of real-valued irreducible characters of finite reductive groups. In particular, if G is a connected reductive group defined over F-q, and psi is an irreducible character of G(F-qm) which is the lift of an irreducible character chi of G(F-q), we prove psi is real-valued if and only if chi is real-valued. In the case m = 2, we show that if chi is invariant under the twisting operator of G(F-q2), and is a real-valued irreducible character in the image of lifting from G(F-q), then. must be an orthogonal character. We also study properties of the Frobenius-Schur indicator under Shintani lifting of regular, semisimple, and irreducible Deligne-Lusztig characters of finite reductive groups.