发表机构
Institut Élie Cartan de Lorraine（洛林埃利嘉当研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对p进SL(n)的tempered对偶的基点问题，构造了n≥8且非素数、非9时W_𝒪无不动点的轨道族，同时证明素数或小维数下W_𝒪总有不动点。
AI 中文摘要
设F是特征为零的非阿基米德局部域，M是G=SL(n,F)的一个Levi子群。考虑M的酉非分歧特征群对M的离散级数的作用（特征通过扭转作用）。对于该作用的一个轨道𝒪，记W_𝒪为𝒪在M的外尔群中的整体稳定子群。我们构造了W_𝒪作用没有不动点的轨道族𝒪。更精确地说，设n≥8且n既非素数也不等于9，我们将无不动点的轨道族与n的任意满足m|q−1的因子m≥2关联起来，其中q是F的剩余域的阶。对应族的Levi子群M⊂G有m个大小为k≥2的块和2个大小为m的块。本文的核心是构造GL(m,F)和GL(k,F)的超尖点表示，这些表示满足构造所需轨道族𝒪的某些条件。此外，我们研究了上述排除的维数（即素数或小维数）下的不动点问题，证明这些情形下W_𝒪作用下总有一个不动点。
英文摘要
Let $F$ be a non-archimedean local field of characteristic zero. Let $M$ be a Levi subgroup of $G = \mathrm{SL}(n,F)$. Consider the action on the discrete series of $M$ of the group of unitary unramified characters of $M$ (where a character acts by twisting). Given an orbit $\mathcal{O}$ for that action, let $W_\mathcal{O}$ be the global stabilizer of $\mathcal{O}$ in the Weyl group of $M$. We construct families of orbits $\mathcal{O}$ for which the action of $W_\mathcal{O}$ has no fixed point. More precisely, let $n \geq 8$ be an integer that is not prime and not equal to $9$. We attach families of orbits without fixed points to any divisor $m \geq 2$ of $n$ such that $m \;| \; q-1$ where $q$ is the order of the residue field of $F$. The Levi subgroup $M \subset G$ for the corresponding families has $m$ blocks of size $k\geq 2$ and $2$ blocks of size $m$. The core of the paper is the construction of superculpidal representations of $\mathrm{GL}(m,F)$ and $\mathrm{GL}(k,F)$ satisfying some conditions that we require to construct the families of orbits $\mathcal{O}$ that we expects. In addition, we study the fixed point problem for the dimensions $n$ that we excluded above, i.e., prime or small dimensions. We prove that there is always a fixed point under the action of $W_\mathcal{O}$ in those cases.
