Local distinction, quadratic base change and automorphic induction for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub></mml:math>

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 ′ a non Archimedean local field F, further translated in terms GL n (F) via Langlands correspondence when has characteristic zero, it yields various statements concerning behaviour different types distinction under quadratic base change automorphic induction. residual from 2, combining one simple results that we obtain with tiviality conjugate-orthogonal root numbers ([8]), recover without using LLC result Serre parity Artin conductor orthogonal ([23]). On other hand discuss its symplectic Prasad Takloo-Bighash conjecture.