论文标题
SL（n）的L-actets内部的区别
Distinction inside L-packets of SL(n)
论文作者
论文摘要
If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N(E)/N(F)$.然后，我们建立了这个结果的全球类似物。为此，令$ e/f $为数字字段的二次扩展，让$π$为$ \ mathrm {sl} _n（\ Mathbb {a} _f）$ - 划分的Square Square-square-square-Antegrable Antegrable Automorphic Automorphic Automorphic Automorphic表示$ \ Mathrm {SL} _N（SL} _n（\ Mathbbbbbbbbbbbbbb {\ Mathbb {a} a} =）令$（σ，d）$是与$π$相关的唯一对，其中$σ$是$ \ mathrm {gl} _r（\ Mathbb {a} _e）$的cuspidal表示，与$ n = dr $。使用一个不断增长的论点，我们证明了$π$的l-packet的一个要素相对于$ \ mathrm {sl} _n（\ Mathbb {a} _f）$，并且仅当它具有变性的惠特克模型的模型时，即$ r^d：=（r^d：=（r^dementere whittaker）模型$ n_n（\ mathbb {a} _e）$，在$ n_n（e+\ mathbb {a} _f）$上是微不足道的，其中$ n_n $是$ \ m mathrm {sl} _n $ $ n_n $的$ n_n $。作为第一个应用程序，在假设$ e/f $在infinity和$ r $上划分的假设是奇怪的，我们为$ \ mathrm {sl} _n（\ Mathbb {a} _f）$ - 在$π$的L-packet内部建立了local-lobal原理。作为第二个应用程序，我们构建了杰出的cuspidal自动形态表示的示例$π$ of $ \ mathrm {sl} _n（\ Mathbb {a} _e）$ $ \ mathrm {sl} _n（\ Mathbb {a} _e）$，使其L-packets不包含任何明显的表示。
If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N(E)/N(F)$. Then we establish a global analogue of this result. For this, let $E/F$ be a quadratic extension of number fields and let $π$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished square integrable automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Let $(σ,d)$ be the unique pair associated to $π$, where $σ$ is a cuspidal representation of $\mathrm{GL}_r(\mathbb{A}_E)$ with $n=dr$. Using an unfolding argument, we prove that an element of the L-packet of $π$ is distinguished with respect to $\mathrm{SL}_n(\mathbb{A}_F)$ if and only if it has a degenerate Whittaker model for a degenerate character $ψ$ of type $r^d:=(r,\dots,r)$ of $N_n(\mathbb{A}_E)$ which is trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $\mathrm{SL}_n$. As a first application, under the assumptions that $E/F$ splits at infinity and $r$ is odd, we establish a local-global principle for $\mathrm{SL}_n(\mathbb{A}_F)$-distinction inside the L-packet of $π$. As a second application we construct examples of distinguished cuspidal automorphic representations $π$ of $\mathrm{SL}_n(\mathbb{A}_E)$ such that the period integral vanishes on some canonical copy of $π$, and of everywhere locally distinguished representations of $\mathrm{SL}_n(\mathbb{A}_E)$ such that their L-packets do not contain any distinguished representation.
