Local distinction, quadratic base change and automorphic induction for GL n

Abstract

Behind this sophisticated title hides an elementary exercise on Clifford theory for index two subgroups and self-dual or conjugate-dual representations. When applied to semi-simple representations of the Weil–Deligne group W F ′ of a non Archimedean local field F, and further translated in terms of representations of GL n (F) via the local Langlands correspondence when F has characteristic zero, it yields various statements concerning the behaviour of different types of distinction under quadratic base change and automorphic induction. When F has residual characteristic different from 2, combining of one of the simple results that we obtain with the tiviality of conjugate-orthogonal root numbers ([8]), we recover without using the LLC a result of Serre on the parity of the Artin conductor of orthogonal representations of W F ′ ([23]). On the other hand we discuss its parity for symplectic representations using the LLC and the Prasad and Takloo-Bighash conjecture.