Distinguished representations for \(\mathrm{SL}(n)\) (Q2424316)

Let \(E/F\) be a quadratic extension of local fields. Let \(\pi\) be an irreducible admissible generic representation of \(\mathrm{SL}(n,E)\). The dimension of \(\mathrm{Hom}_{\mathrm{SL}(n,F)}(\pi,\mathbb{C})\) is computed and is related to fibers of the base change map of representations from \(\mathrm{SU}(n,F)\) to \(\mathrm{SL}(n,E)\) as suggested in a previous paper of the second author [Compos. Math. 119, No. 3, 335--345 (1999; Zbl 0969.22008)]. Finite fields are considered too.