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 $D/F$ to that of its L-parameter $\varphi_\sigma\colon W_F\to \mathrm{GL}_n(\mathbb C)$, extending results of [Prasad-Ramakrishnan]. The field of definitions are controlled by division algebras $\mathcal D_{\sigma}$ and $\mathcal D_{\varphi_\sigma}$ 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 $p$.