Shintani lifting and real-valued characters

Abstract

We study Shintani lifting of real-valued irreducible characters of finite reductive groups. In particular, if G is a connected reductive group defined over \({\mathbb{F}_q}\), and ψ is an irreducible character of G(\({\mathbb{F}_{q^m}}\)) which is the lift of an irreducible character χ of G(\({\mathbb{F}_q}\)), 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(\({\mathbb{F}_{q^2}}\)), and is a real-valued irreducible character in the image of lifting from G(\({\mathbb{F}_q}\)), 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.