Distinguished representations for SL(n)

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