Idea of base change for Division algebras over local field

Question

Let $F$ be a non-Archimedean local field of characteristic $0$ and $K/F$ be a finite extension. Let $D_F$ be the central division algebra of dimension $n^2$ over $F.$ Write $D_K=D_F\otimes_FK$, which is again a central division algebra over $K$ of dimension $n^2$. Does there exist an idea of base change for division algebra.

In case of $GL(n),$ the following diagram is commutative.

$$ \matrix{ \widehat{W_F} & \longrightarrow^{{}^{LLoc}} &\widehat{ GL(n,F)} \cr \downarrow^{res_{K/F}}& & \downarrow^{BC_{K/F}} \cr \widehat{W_K} & \longrightarrow^{{}^{LLoc}} & \widehat{GL(n,K)} \cr } $$ where LLoc is Local Langlands correspondence, $res_{K/F}$ is restriction map and $BC_{K/F}$ is base change map.

Do we have similar diagram in case of Division algebras. If yes, how the base change map look like ?

More generally,

$A_F$ be the finite central simple algebra over $F$, which is isomorphic to $M_n(D)$ for some division algebra $D$(unique upto isomorphism) over $F$ of index $d$. Set $A_K=A_F\otimes_FK.$

Do we have similar commutative diagram in case of central simple algebras as above. If the answer is affirmative,suggest some reference regarding this..? Thank you.

Answer 1

An obvious idea to get a commutative square diagram, where $GL(n,F)$ and $GL(n,K)$ are respectively replaced by $D_F^\times$ and $D_K^\times$, is to use the Jacquet-Langlands transfer between representations of $D_F^\times$ (resp. $D_K^\times$) and square integrable representations of $GL(n,F)$ (resp. $GL(n,K)$) (cf. works of Deligne-Kazdhan-Vignéras, Rogawski, Badulescu). Of course this is not satisfactory for base change does not preserve square-integrability in general. But you could may be use the extension of the Jacquet-Langlands transfer to all representations, written by Badulescu.