  For $E/F$ a quadratic extension of local fields, and $\pi$ an irreducible
admissible generic representation of $SL_n(E)$, we calculate the dimension of
$Hom_{SL_n(F)}[\pi,C]$ and relate it to fibers of the base change map
corresponding to base change of representations of $SU_n(F)$ to $SL_n(E)$ as
suggested in a recent work of the second author. We also deal with finite
fields.
