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Behind this sophisticated title hides an elementary exercise on Cliffordtheory for index two subgroups and self-dual/conjugate-dual representations.When applied to semi-simple representations of the Weil-Deligne group $W'_F$ ofa non Archimedean local field $F$, and further translated in terms ofrepresentations of $\mathrm{GL_n}(F)$ via the local Langlands correspondencewhen $F$ has characteristic zero, it yields various statements concerning thebehaviour of different types of distinction under quadratic base change andautomorphic induction. When $F$ has residual characteristic different from $2$,combining of one of the simple results that we obtain with the tiviality ofconjugate-orthogonal root numbers (proved by Gan, Gross and Prasad), we recoverwithout using the LLC a result of Serre on the parity of the Artin conductor oforthogonal representations of $W'_F$. On the other hand we discuss its parityfor symplectic representations using the LLC and the Prasad and Takloo-Bighashconjecture.

Local distinction, quadratic base change and automorphic induction for $\mathrm{GL_n}$