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

TL;DR

This paper explores how distinction properties of representations behave under quadratic base change and automorphic induction, using elementary Clifford theory and the local Langlands correspondence, with implications for Artin conductors and root numbers.

Contribution

It provides new elementary proofs and insights into the behavior of distinction under base change and induction, connecting Clifford theory with the local Langlands correspondence.

Findings

Recovered Serre's parity result for orthogonal representations without LLC

Discussed parity of symplectic representations using LLC and conjectures

Connected distinction properties with Artin conductors and root numbers

Abstract

Behind this sophisticated title hides an elementary exercise on Clifford theory for index two subgroups and self-dual/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 $\mathrm{GL_n}(F)$ 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 (proved by Gan, Gross and Prasad), we recover without using the LLC a result of Serre on the parity of the Artin conductor of orthogonal representations of $W'_F$. On the other hand we discuss its parity for symplectic representations using the LLC and the Prasad and Takloo-Bighash conjecture.