$\mathrm{GL_n}$的局部判别、二次基变换和自守归纳
Local distinction, quadratic base change and automorphic induction for $\mathrm{GL_n}$
摘要：
在这个复杂的标题背后隐藏着一个关于克利福德理论的基本练习，用于指数2子群和自对偶/共轭对偶表示。当应用于非阿基米德局部域$F$的Weil-Deligne群$W'_F$的半简单表示时，并进一步通过局部朗兰兹对应将其转换为$\mathrm{GL_n}(F)$的表示，当$F$具有特征零时，它产生了关于二次基变换和自守归纳下不同类型区别的行为的各种陈述。当$F$具有不同于$2$的剩余特征时，结合我们得到的一个简单结果与共轭正交根数的有效性（GAN，Gross和Prasad证明的），我们在不使用LLC的情况下恢复了Serre关于$W'_F$正交表示的Artin导体奇偶性的结果.另一方面，我们讨论了它的奇偶性辛表示使用LLC和Prasad和Takloo-Bighash猜想。
英文摘要：
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.
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