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

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 ac