Rationality of the Local Jacquet-Langlands Correspondence for GL(n)

We relate the field of definition of representations $\sigma$ of the group of units $D^{\times }$ of a non-archimedean division algebra $D\mathrm{/}F$ to that of its L-parameter ${\phi }_{\sigma }\text{\hspace{0.17em}⁣}:W_F\to {\mathrm{G}\mathrm{L}}_n(\mathbb{C})$, extending results of [Prasad-Ramakrishnan]. The field of definitions are controlled by division algebras ${\mathcal{D}}_{\sigma }$ and ${\mathcal{D}}_{{\phi }_{\sigma }}$ over the field of rationality $\mathbb{Q}(\pi )$, and we completely pin down the relationship between the Hasse invariants at places not over $p$. Under some additional assumptions we can also specify the Hasse invariants at places over $p$.