On a basepoint issue in the tempered dual of $p$-adic $SL(n)$

Abstract

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.