The split case of the Prasad--Takloo-Bighash conjecture for cuspidal representations of level zero

Distinction of the Steinberg representation and a conjecture of Prasad and Takloo-Bighash

We prove a conjecture of Prasad and Takloo-Bighash for discrete series of inner forms of the general linear group over a non archimedean local field, in the case of Steinberg representations.

Prasad与Takloo-Bighash猜想的重新表述

A reformulation of the conjecture of Prasad and Takloo-Bighash

本文将Prasad和Takloo-Bighash关于表示具有线性周期必要条件的猜想，重新表述为泛型L-packet中的区分成员由分量群特征和局部epsilon因子决定，并指出在特定假设下可由相关多重性与积分公式推导得出。

Multiplicity one for pairs of Prasad--Takloo-Bighash type

Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let $A$ be an $F$-central simple algebra of even dimension so that it contains $E$ as a subfield, set $G=A^\times$ and $H$ for the centralizer of $E^\times$ in $G$. Using a Galois descent argument, we prove that all double cosets $H g H\subset G$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo's result for $F$-split $G$ which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that $H$-distinguished irr…

GL(2n)的中心L值的周期与非消失性

Periods and nonvanishing of central L-values for GL(2n)

本文证明Guo-Jacquet关于GL(2n)中心L值非消失性与GL(n,E)周期非消失性关系的猜想，并应用于超尖点表示，部分证实Prasad-Takloo-Bighash猜想。

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

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…

线性模型的 epsilon 二分法：Archimedees 情形

Epsilon dichotomy for linear models: the Archimedean case

本文研究实或四元数一般线性群中 $χ_H$-区分表示的根数，通过旗流形 $H$-轨道与 Schwartz 同调分析证明 Prasad–Takloo-Bighash 猜想的 Archimedees 版本。