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Distinguished representations for SL(n)

For $E/F$ a quadratic extension of local fields, and $π$ an irreducible admissible generic representation of $SL_n(E)$, we calculate the dimension of $Hom_{SL_n(F)}[π,C]$ 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.