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 Fq, and ψ is an irreducible character of G(Fqm) which is the lift of an irreducible character χ of G(Fq), we prove ψ is real-valued if and only if χ is real-valued. In the case m = 2, we show that if χ is invariant under the twisting operator of G(Fq2), and is a real-valued irreducible character in the image of lifting from G(Fq), then χ 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.