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.



Idea of base change for Division algebras over local field
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. 


